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FAMOUS  PBOBLEMS 


ELEMENTARY    GEOMETRY 


THE  DUPLICATION   OF  THE   CUBE 
THE  TRISECTION   OF   AN  ANGLE 
THE  QUADRATURE  OF  THE  CIRCLE 


A2f  AUTHORIZED  TEAIfSLATIOK  OF  F.   KLEIN'S 

VOBTBAGE  UBER  AUSGEWAHLTE  FKAGEN  DER  ELEMENTARGEOMETRIE 

AUSGEARBEITET  VON  F.  TAGERT 


BT 

WOOSTER   WOODRUFF    BEMAN 

Professor  of  Mathematics  in  the  Ukiversitv  of  Michigan 

AXD 

DAVID    EUGENE   SMITH 

Pkokk^mjk  01  Matiiematus  in  Teachers  CoLLECii,  Colimbia  Umversiiv 


GINN  AND  COxMPANY 

noSTON     •     NKW    YOIiK     ■     CHICAGO     ■     I.ONKON 
ATLANTA     •     DALLAS     •     COLfMKCS     •    SAX    KJtAXCISCO 

3  7H^. 


COPYKIGHT,  1897,  BY 
WOOSTBB  WOODKUFF  BEMAlf  AND  DAVID  EUOENE  SBOTH 


Alili  BIGHTS  BESEKYBD 
316.9 


tCftt   att)tngum   jgregg 

GINN  AND  COMPANY-  PRO- 
PRIETORS ■  POSTON  •  U,S.A. 


Q?  A\  Sci-nces 


PREFACE. 


The  more  precise  definitions  and  more  rigorous  methods  of 
demonstration  developed  by  modern  mathematics  are  looked 
upon  by  the  mass  of  gymnasium  professors  as  abstruse  and 
excessively  abstract,  and  accordingly  as  of  importance  only 
for  the  small  circle  of  specialists.  With  a  view  to  counteract- 
ing this  tendency  it  gave  me  pleasure  to  set  forth  last  summer 
in  a  brief  course  of  lectures  before  a  larger  audience  than 
[  usual  what  modern  science  has  to  say  regarding  the  possibility 

*  of  elementary  geometric  constructions.     Some  time  before,  I 
^  had  had  occasion  to  present  a  sketch  of  these  lectures  in  an 

*  Easter  vacation  course  at  Gottingen.     The  audience  seemed 
to  tak6  great  interest  in  them,  and  this  impression  has  been 

I  confirmed  by  the  experience  of  the  summer  semester.  I  ven- 
\  ture  therefore  to  present  a  short  exposition  of  my  lectures  to 
the  Association  for  the  Advancement  of  the  Teaching  of  Math- 
ematics and  the  Natural  Sciences,  for  the  meeting  to  be  held  at 
Gottingen.  This  exposition  has  been  prepared  by  Oberlehrer 
Tagert,  of  Ems,  who  attended  the  vacation  course  just  men- 
tioned. He  also  had  at  his  disposal  the  lecture  notes  written 
out  under  my  supervision  by  several  of  my  summer  semester 
students.  I  hope  that  this  unpretending  little  book  may  con- 
tribute to  promote  the  useful  work  of  the  association. 

F.  KLEIN. 
CiOTTiNOEN,  Easter,  1895. 


TRANSLATORS^    PREFACE. 


At  the  Gottingen  meeting  of  the  German  Association  for 
the  Advancement  of  the  Teaching  of  Mathematics  and  the 
Natural  Sciences,  Professor  Pelix  Klein  presented  a  discus- 
sion of  the  three  famous  geometric  problems  of  antiquity, 
—  the  duplication  of  the  cube,  the  trisection  of  an  angle, 
and  the  quadrature  of  the  circle,  as  viewed  in  the  light  of 
modern  research. 

This  was  done  with  the  avowed  purpose  of  bringing  the 
study  of  mathematics  in  the  university  into  closer  touch  with 
the  work  of  the  gymnasium.  That  Professor  Klein  is  likely 
to  succeed  in  this  effort  is  shown  by  the  favorable  reception 
accorded  his  lectures  by  the  association,  the  uniform  commen- 
dation of  the  educational  journals,  and  the  fact  that  transla- 
tions into  French  and  Italian  have  already  appeared. 

The  treatment  of  the  subject  is  elementary,  not  even  a 
knowledge  of  the  differential  and  integral  calculus  being 
required.  Among  the  questions  answered  are  such  as  these  : 
Under  what  circumstances  is  a  geometric  construction  pos- 
sible ?  By  what  means  can  it  be  effected  ?  What  are  tran- 
scendental numbers?  How  can  we  prove  that  e  and  ir  are 
transcendental  ? 

With  the  belief  that  an  English  presentation  of  so  impor- 
tant a  work  would  appeal  to  many  unable  to  read  the  original, 


Vi  TRANSLATOR'S  PREFACE. 

Professor  Klein's  consent  to  a  translation  was  sought  and 
readily  secured. 

In  its  preparation  the  authors  have  also  made  free  use  of 
the  French  translation  by  Professor  J.  Griess,  of  Algiers, 
following  its  modifications  where  it  seemed  advisable. 

They  desire  further  to  thank  Professor  Ziwet  for  assist- 
ance in  improving  the  translation  and  in  reading  the  proof- 
sheets. 


August,  1897. 


W.  W.  BEMAN. 
D.  E.  SMITH. 


CONTENTS. 


INTRODUCTION. 

PAOB 

Practical  and  Theoretical  Constructions  ....       2 

Statement  of  the  Problem  in  Algebraic  Form       ...  3 

PART   I. 

The  Possibility  of  the  Construction  of  Algebraic  Expressions. 

Chapter  I.     Algebraic    Equations    Solvable   by   Square   Roots. 

1-4.  Structure  of  the  expression  x  to  be  constructed      ...  5 

5,  6.  Normal  form  of  x 6 

7,  8.  Conjugate  values 7 

9.  The  corresponding  equation  F{x)  =  o          ....  8 

10.  Other  rational  equations  f(x)  =  o 8 

11,  12.  The  irreducible  equation  <t>{x)  =  o 10 

13,  14.  The  degree  of  the  irreducible  equation  a  power  of  2      .         .11 

Chapter  II.     The  Delian   Problem  and  the  Trisection  of   the 

Angle. 

1.  The  impossibility  of  solving  the  Delian  problem  with  straight 

edge  and  compasses 13 

2.  The  general  equation  x^  =  X 13 

3.  The  impossibility  of  trisecting  an  angle  with  straight  edge 

and  compasses 14 

Chapter  III.     The   Division   of   the   Circle   into   Equal   Parts. 

1.  History  of  the  problem 16 

2-4.  Gauss's  prime  numbers 17 

5.  The  cyclotomic  equation 19 

6.  Gauss's  Lemma 19 

7,  8.  The  irreducibility  of  the  cyclotomic  equation         .        .        .21 


viii  CONTENTS. 

Chapter   IV.     The    Construction   of   the   Regular   Polygon   of 
17  Sides. 

PAOB 

1.  Algebraic  statement  of  the  problem 24 

2-4.  The  periods  formed  from  the  roots 25 

6,  6.  The  quadratic  equations  satisfied  by  the  periods        .         .  27 

7.  Historical  account  of  constructions  with  straight  edge  and 

compasses 32 

8,  9.  Von  Staudt's  construction  of  the  regular  polygon  of  17  sides  34 

Chapter  V.    General  Considerations  on  Algebraic  Constructions. 

1.  Paper  folding 42 

2.  The  conic  sections 42 

3.  The  Cissoid  of  Diodes 44 

4.  The  Conchoid  of  Nicomedes 45 

5.  Mechanical  devices         ........  47 


PART  II. 

Transcendental  Numbers  and  the  Quadrature  of  the  Circle. 

Chapter    I.      Cantor's    Demonstration    of    the    Existence    of 
Transcendental  Numbers. 

1.  Definition  of  algebraic  and  of  transcendental  numbers       .        49 

2.  Arrangement  of  algebraic  numbers  according  to  height  .     50 

3.  Demonstration  of  the  existence  of  transcendental  numbers        53 


Chapter  II.     Historical  Survey  of  the  Attempts  at  the  Com- 
putation AND  Construction  of  tt. 

1.  The  empirical  stage 56 

2.  The  Greek  mathematicians 66 

3.  Modern  analysis  from  1670  to  1770 58 

4,  5.  Revival  of  critical  rigor  since  1770 59 

Chapter  III.     The  Transcendence  of  the  Number  e. 

1.  Outline  of  the  demonstration 61 

2.  The  symbol  hr  and  the  function  0(x)  ....  62 

3.  Hermite's  Theorem 65 


CONTENTS.  ix 

Chapter  IV.     The  Tkanscendence  of  the  Number  tt. 

PAOI 

1.  Outline  of  the  demonstration 68 

2.  The  function  ^{x) 70 

3.  Lindemann's  Theorem 73 

4.  Lindemann's  Corollary 74 

5.  The  transcendence  of  tt       .         .         .         .         .         .         .  76 

6.  The  transcendence  of  y  =  e" 77 

7.  The  transcendence  of  y  =  sin-^x 77 

Chapter  V.     The   Integraph   and   the   Geometric    Construction 

OF    T. 

1.  The  impossibility  of  the  quadrature  of  the  circle  with  straight 

edge  and  compasses 78 

2.  Principle  of  the  integraph 78 

3.  Geometric  construction  of  tt  .        .        .        .        .        .        .79 


INTRODUCTION. 


This  course  of  lectures  is  due  to  the  desire  on  my  part  to 
bring  the  study  of  mathematics  in  the  university  into  closer 
touch  with  the  needs  of  the  secondary  schools.  Still  it  is  not 
intended  for  beginners,  since  the  matters  under  discussion  are 
treated  from  a  higher  standpoint  than  that  of  the  schools. 
On  the  other  hand,  it  presupposes  but  little  preliminary  work, 
only  the  elements  of  analysis  being  required,  as,  for  example, 
in  the  development  of  the  exponential  function  into  a  series. 

We  propose  to  treat  of  geometrical  constructions,  and  our 
object  will  not  be  so  much  to  find  the  solution  suited  to  each 
case  as  to  determine  the  possibility  or  impossibility  of  a 
solution. 

Three  problems,  the  object  of  much  research  in  ancient 
times,  will  prove  to  be  of  special  interest.     They  are 

1.  The  problem  of  the  duplication  of  the  cube  (also  called 
the  Delian  problevi). 

2.  The  trisection  of  an  arbitrary  angle. 

3.  The  quadrature  of  the  circle,  i.e.,  the  construction  of  ir. 

In  all  these  problems  the  ancients  sought  in  vain  for  a 
solution  with  straight  edge  and  compasses,  and  the  celebrity 
of  these  problems  is  due  chiefly  to  the  fact  that  their  solution 
seemed  to  demand  the  use  of  appliances  of  a  higher  order. 
In  fact,  we  propose  to  show  that  a  solution  by  the  use  of 
straight  edge  and  compasses  is  impossible. 


2  INTRODUCTION. 

The  impossibility  of  the  solution  of  the  third  problem  was 
demonstrated  only  very  recently.  That  of  the  first  and  second 
is  implicitly  involved  in  the  Galois  theory  as  presented  to-day 
in  treatises  on  higher  algebra.  On  the  other  hand,  we  find 
no  explicit  demonstration  in  elementary  form  unless  it  be  in 
Petersen's  text-books,  works  which  are  also  noteworthy  in 
other  respects. 

At  the  outset  we  must  insist  upon  the  difference  between 
practical  and  theoretical  constructions.  For  example,  if  we 
need  a  divided  circle  as  a  measuring  instrument,  we  construct 
it  simply  on  trial.  Theoretically,  in  earlier  times,  it  was 
possible  (i.e.,  by  the  use  of  straight  edge  and  compasses)  only 
to  divide  the  circle  into  a  number  of  parts  represented  by 
2",  3,  and  5,  and  their  products.  Gauss  added  other  cases 
by  showing  the  possibility  of  the  division  into  parts  where 
p  is  a  prime  number  of  the  form  p  =  2^*"  -}-  1,  and  the  impos- 
sibility for  all  other  numbers.  No  practical  advantage  is 
derived  from  these  results;  the  significance  of  Gauss's  de- 
velopments is  purely  theoretical.  The  same  is  true  of  all  the 
discussions  of  the  present  course. 

Our  fundamental  problem  may  be  stated  :  What  geometrical 
constructions  are,  and  what  are  not,  theoretically  possible  ?  To 
define  sharply  the  meaning  of  the  word  "construction,"  we 
must  designate  the  instruments  which  we  propose  to  use  in 
each  case.     We  shall  consider 

1.  Straight  edge  and  compasses, 

2.  Compasses  alone, 

3.  Straight  edge  alone, 

4.  Other  instruments  used  in  connection  with  straight  edge 
and  compasses. 

The  singular  thing  is  that  elementary  geometry  furnishes 
no  answer  to  the  question.  We  must  fall  back  upon  algebra 
and  the  higher  analysis.      The  question  then  arises :   How 


INTRODUCTION.  3 

shall  we  use  the  language  of  these  sciences  to  express  the 
employment  of  straight  edge  and  compasses  ?  This  new- 
method  of  attack  is  rendered  necessary  because  elementary 
geometry  possesses  no  general  method,  no  algorithm,  as  do 
the  last  two  sciences. 

In  analysis  we  have  first  rational  operations :  addition, 
subtraction,  multiplication,  and  division.  These  operations 
can  be  directly  effected  geometrically  upon  two  given  seg- 
ments by  the  aid  of  proportions,  if,  in  the  case  of  multiplica- 
tion and  division,  we  introduce  an  auxiliar}^  unit-segment. 

Further,  there  are  irrational  operations,  subdivided  into 
algebraic  and  transcendental.  The  simplest  algebraic  opera- 
tions are  the  extraction  of  square  and  higher  roots,  and  the 
solution  of  algebraic  equations  not  solvable  by  radicals,  such 
as  those  of  the  fifth  and  higher  degrees.  As  we  know  how  to 
construct  Vab,  rational  operations  in  general,  and  irrational 
operations  involving  only  square  roots,  can  be  constructed. 
On  the  other  hand,  every  individual  geometrical  construction 
which  can  be  reduced  to  the  intersection  of  two  straight 
lines,  a  straight  line  and  a  circle,  or  two  circles,  is  equivalent 
to  a  rational  operation  or  the  extraction  of  a  square  root.  In 
the  higher  irrational  operations  the  construction  is  therefore 
impossible,  miless  we  can  find  a  way  of  effecting  it  by  the  aid 
of  square  roots.  In  all  these  constructions  it  is*  obvious  that 
the  number  of  operations  must  be  limited. 

We  may  therefore  state  the  following  fundamental  theorem : 
The  necessary  and  sufficient  condition  that  an  analytic  expres- 
sion can  be  constructed  ivith  straight  edge  and  compasses  is  that 
it  can  be  derived  from  the  known  quantities  by  a  finite  nwtnber 
of  rational  operations  and  square  roots. 

Accordingly,  if  we  wish  to  show  that  a  quantity  cannot  be 
constructed  with  straight  edge  and  compasses,  we  must  prove 
that  the  corresponding  equation  is  not  solvable  by  a  finite 
number  of  square  roots. 


4  INTRODUCTION. 

A  fortiori  the  solution  is  impossible  when  the  problem 
has  no  corresponding  algebraic  equation.  An  expression 
which  satisfies  no  algebraic  equation  is  called  a  transcenden- 
tal number.  This  case  occurs,  as  we  shall  show,  with  the 
number  tt. 


PART   I. 


THE    POSSIBILITY    OF    THE    CONSTRUCTION    OF    ALGEBBAIC 
EXFBESSIONS. 


CHAPTER   I. 

Algebraic  Equations  Solvable  by  Square  Roots. 

The  following  propositions  taken  from  tlie  theory  of  alge- 
braic equations  ar*  probably  known  to  the  reader,  yet  to 
secure  greater  clearness  of  view  we  shall  give  brief  demon- 
strations. 

If  X,  the  quantity  to  be  constructed,  depends  only  upon  rational 
expressions  and  square  roots,  it  is  a  root  of  an  irreducible  equa- 
tion f  (x)  =  0,  whose  degree  is  always  a  power  of  2. 

1.  To  get  a  clear  idea  of  the  structure  of  the  quantity  x, 
suppose  it,  e.g.,  of  the  form 

^  ^  Va  +  Vc  +  ef  +  Vd^+  Vb      p_+Vq^ 
Va  +  Vb  V7     ' 

where  a,  b,  c,  d,  e,  f,  p,  q,  r  are  rational  expressions. 

2.  The  number  of  radicals  one  over  another  occurring  in 
any  term  of  x  is  called  the  order  of  the  term  ;  the  preceding 
expression  contains  terms  of  orders  0,  1,  2. 

3.  Let  fjL  designate  the  maximum  order,  so  that  no  term 
can  have  more  than  /*  radicals  one  over  another. 


6  FAMOUS  PROBLEMS. 

4.  In  the  example  x  =  \/2  +  V3  +  V6,  we  have  three 
expressions  of  the  first  order,  but  as  it  may  be  written 

x  =  V2+ V3+ V2- V3, 

it  really  depends  on  only  two  distinct  expressions. 

We  shall  suppose  that  this  reduction  has  been  made  in  all  the 
terms  of  x,  so  that  arnong  the  n  terms  of  order  fx  none  can  be 
exjjressed  rationally  as  a  function  of  any  other  terms  of  order  /j. 
or  of  lower  order. 

We  shall  make  the  same  supposition  regarding  terms  of 
the  order  fi  —  1  or  of  lower  order,  whether  these  occur  ex- 
plicitly or  implicitly.  This  hypothesis  is  obviously  a  very 
natural  one  and  of  great  importance  in  later  discussions. 

5.  Normal  Form  of  x. 

If  the  expression  x  is  a  sum  of  terms  with  different  denom- 
inators we  may  reduce  them  to  the  same  denominator  and 
thus  obtain  x  as  the  quotient  of  two  integral  functions. 

Suppose  VQ  one  of  the  terms  of  x  of  order  fi ;  it  can  occur 
in  X  only  explicitly,  since  /x.  is  the  maximum  order.  Since, 
further,  the  powers  of  VQ  may  be  expressed  as  functions  of 
VQ  and  Q,  which  is  a  term  of  lower  order,  we  may  put 

^  a  +  b  VQ 
c-fdVQ' 
where  a,  b,  c,  d  contain  no  more  than  n  —  1  terms  of  order  fi, 
besides  terms  of  lower  order. 

Multiplying  both  terms  of  the  fraction  by  c  —  d  VQ,  VQ 
disappears  from  the  denominator,  and  we  may  write 

(ac-bdQ)+(bc-ad)VQ_  /^ 

where  a  and  /3  contain  no  more  than  n  —  1  terms  of  order  fi. 

For  a  second  term  of  order  fi  we  have,  in  a  similar  manner, 
X  =  tti  +  )8i  VQi,  etc. 


ALGEBRAIC  EQUATIONS.  7 

The  X  may,  therefore,  be  transformed  so  as  to  contain  a  term 
of  given  order  yn  only  in  its  numerator  and  there  only  linearly. 

We  observe,  however,  that  products  of  terms  of  order  /x 
may  occur,  for  a  and  ft  still  depend  upon  n  —  1  terms  of  order 
fx.     We  may,  then,  put 

a  =  a„  +  ai2  VQi,  )8  =  Ai  +  l^u  VQi, 

and  hence 

X  =  (a„  +  ai2  VQ^)  +  (/3n  +  /812  VQi)  VQ. 

6.  We  proceed  in  a  similar  way  with  the  different  terms 
of  order  /u,  —  1,  which  occur  explicitly  and  in  Q,  Q^,  etc.,  so 
that  each  of  these  quantities  becomes  an  integral  linear  func- 
tion of  the  term  of  order  fx.  —  1  under  consideration.  We 
then  pass  on  to  terms  of  lower  order  and  finally  obtain  x,  or 
rather  its  terms  of  different  orders,  under  the  form  of  rational 
integral  linear  functions  of  the  individual  radical  expressions 
which  occur  explicitly.  We  then  say  that  x  is  reduced  to 
the  normal  form. 

7.  Let  m  be  the  total  number  of  independent  (4)  square 
roots  occurring  in  this  normal  form.  Giving  the  double  sign 
to  these  square  roots  and  combining  them  in  all  possible  ways, 
we  obtain  a  system  of  2"*  values 

Xi,    X2,   ....    X^jjj, 

which  we  shall  call  conjugate  values. 

We  must  now  investigate  the  equation  admitting  these 
conjugate  values  as  roots. 

8.  These  values  are  not  necessarily  all  distinct ;  thus,  if 


we  have  x  =  \J di  +  Vb  -|-  \/a  —  Vb, 

this  expression  is  not  changed  when  we  change  the  sign  of 

Vb. 


8  FAMOUS  PROBLEMS. 

9.  If  X  is  an  arbitrary  quantity  and  we  form  the  poly- 
nomial 

F  (x)  =  (X  —  xi)  (x  —  X2)  .  .  .  (x  —  x^J, 

F  (x)  =  0  is  clearly  an  equation  having  as  roots  these  con- 
jugate values.  It  is  of  degree  2™,  but  may  have  equal 
roots  (8). 

The  coefficients  of  the  polynomial  F  (x)  arranged  with  respect 
to  x  are  rational. 

For  let  us  change  the  sign  of  one  of  the  square  roots  ;  this 
will  permute  two  roots,  say  x^  and  x^',  since  the  roots  of 
F  (x)  =:  0  are  precisely  all  the  conjugate  values.  As  these 
roots  enter  F  (x)  only  under  the  form  of  the  product 

(X  —  Xa)  (x  —  Xx'), 

we  merely  change  the  order  of  the  factors  of  F  (x).  Hence 
the  polynomial  is  not  changed. 

F  (x)  remains,  then,  invariable  when  we  change  the  sign  of 
any  one  of  the  square  roots  ;  it  therefore  contains  only  their 
squares  ;  and  hence  F  (x)  has  only  rational  coefficients. 

,  10.  When  any  one  of  the  conjugate  values  satisfies  a  given 
equation  with  rational  coefficients,  f  (x)  =  0,  the  same  is  true  of 
all  the  others. 

f  (x)  is  not  necessarily  equal  to  F  (x),  and  may  admit  other 
roots  besides  the  x/s. 

Let  Xi  =  a  +  )8  VQ  be  one  of  the  conjugate  values  ;  VQ,  a 
term  of  order  fx  ;  a  and  yS  now  depend  only  upon  other  terms 
of  order  ix.  and  terms  of  lower  order.  There  must,  then,  be  a 
conjugate  value 

xi'  =  a  — )8VQ. 

Let  us  now  form  the  equation  f  (xj)  =  0.  f  (xi)  may  be  put 
into  the  normal  form  with  respect  to  VQ, 

f(xi)  =  A+BVQ; 


ALGEBRAIC  EQUATIONS.  9 

this  expression  can  equal  zero  only  when  A  and  B  are  simul- 
taneously zero.     Otherwise  we  should  have 

i.e.,  VQ  could  be  expressed  rationally  as  a  function  of  terms 
of  order  fi  and  of  terms  of  lower  order  contained  in  A  and  B, 
which  is  contrary  to  the  hypothesis  of  the  independence  of 
all  the  square  roots  (4). 
But  we  evidently  have 

f(xO  =  A-B  VQ; 

.hence  if  f  (xi)  =  0,  so  also  f  (xj')  =  0.  Whence  the  following 
proposition  : 

If  Xi  satisfies  the  equation  f  (x)  =  0,  the  same  is  true  of  all 
the  conjugate  values  derived  from  Xi  by  changing  the  signs  of 
the  roots  of  order  /a. 

The  proof  for  the  other  conjugate  values  is  obtained  in  an 
analogous  manner.  Suppose,  for  example,  as  may  be  done 
without  affecting  the  generality  of  the  reasoning,  that  the 
expression  x^  depends  on  only  two  terms  of  order  /a,  VQ  and 
VQ'.     f  (xj)  may  be  reduced  to  the  following  normal  form  : 

(a)  f  (x,)  =  p  +  q  VQ  +  r  VQ^  +  s  VQ  •  VQ^=  0. 

If  Xj  depended  on  more  than  two  terms  of  order  /x,  we  should 
only  have  to  add  to  the  preceding  expression  a  greater  num- 
ber of  terms  of  analogous  structure. 

Equation  (a)  is  possible  only  when  we  have  separately 

(h)  p  =  0,     q  =  0,     r  =  0,     s  =  0. 

Otherwise  VQ  and  VQ'  would  be  connected  by  a  rational 
relation,  contrary  to  our  hypothesis. 

Let  now  VR,  VR',  ...  be  the  terms  of  order  ft.  —  1  on 
which  Xi  depends  ;  they  occur  in  p.  q,  r,  s ;  then  can  the 
quantities  p,  q,  r,  s,  in  which  they  occur,  be  reduced  to  the 


10  FAMOUS  PROBLEMS. 

normal  form  with  respect  to  VR  and  VR'  ;  andj^f,  for  the 
sake  of  simplicity,  we  take  only  two  quantities,  VR  and  VR', 
we  have 

(c)  p  =  Ki  +  XxVR  +  ftiVR'  +  viVR.VR'  =  0, 

and  three  analogous  equations  for  q,  r,  s. 

The  hypothesis,  already  used  several  times,  of  the  inde- 
pendence of  the  roots,  furnishes  the  equations 

(d)  K  =  0,     X  =  0,     ^  =  0,     v  =  0. 

Hence  equations  (c)  and  consequently  f  (x)  =  0  are  satisfied 
when  for  Xi  we  substitute  the  conjugate  values  deduced  by 
changing  the  signs  of  VR  and  VR'. 

Therefore  the  equation  f  (x)  =  0  is  also  satisfied  by  all  the 
conjugate  values  deduced  from  Xi  by  changing  the  signs  of  the 
roots  of  order  /x  —  1. 

The  same  reasoning  is  applicable  to  the  terms  of  order 
/A  —  2,    fi  —  3,  .  .  .  and   our    theorem  is  completely    proved. 

1 1 .  We  have  so  far  considered  two  equations 

F  (x)  =  0     and     f  (x)  =  0. 

Both  have  rational  coefficients  and  contain  the  Xj's  as  roots. 
F  (x)  is  of  degree  2"  and  may  have  multiple  roots  ;  f  (x)  may 
have  other  roots  besides  the  Xj's.  We  now  introduce  a  tliird 
equation,  <f>  (x)  =  0,  defined  as  the  equation  of  lowest  degree, 
with  rational  coefficients,  admitting  the  root  Xj  and  conse- 
quently all  the  Xi's  (10). 

1 2.  Properties  of  the  Equation  ^  (x)  =  0. 

I.  <^  (x)  ::^  0  ts  an  irreducible  equation,  i.e.,  <f>  (x)  cannot  be 
resolved  into  two  rational  polynomial  factors.  This  irreduci- 
bility  is  due  to  the  hypothesis  that  <^  (x)  =  0  is  the  rational 
equation  of  lowest  degree  satisfied  by  the  x/s. 

For  if  we  had 

</»W  =  >A(x)x(x), 


ALGEBRAIC  EQUATIONS.  11 

then  (f}  (xx)  =  0  would  require  either  if/  (xi)  =  0,  or  ^  (xi)  =  0, 
or  both.  But  since  these  equations  are  satisfied  by  all  the 
conjugate  values  (10),  ^  (x)  =0  would  not  then  be  the  equa- 
tion of  lowest  degree  satisfied  by  the  x/s. 

II.  <^  (x)  =  0  has  no  multiple  roots.  Otherwise  <f>  (x)  could 
be  decomposed  into  rational  factors  by  the  well-known  meth- 
ods of  Algebra,  and  <^  (x)  =  0  would  not  be  irreducible. 

III.  <^  (x)  ^  0  has  no  other  roots  than  the  x/s.  Otherwise 
F  (x)  and  <;^  (x)  would  admit  a  highest  common  divisor,  which 
could  be  determined  rationally.  We  could  then  decompose 
<i>  (x)  into  rational  factors,  and  <f>  (x)  would  not  be  irreducible. 

IV.  Let  M  be  the  number  of  x/s  which  have  distinct  values, 
and  let 


be  these  quantities.     We  shall  then  have 

«^  (x)  =  C  (x  —  xi)  (x  — .  xa)  .  .  .  (x  —  Xm). 

For  ^  (x)  =  0  is  satisfied  by  the  quantities  Xj  and  it  has  no 
multiple  roots.  The  polynomial  <^  (x)  is  then  determined  save 
for  a  constant  factor  whose  value  has  no  effect  upon  <j>  (x)  =  0. 

V.  <f>  (x)  =0  is  the  only  irreducible  equation  with  rational 
coefficients  satisfied  by  the  x/s.  For  if  f  (x)  =  0  were  another 
rational  irreducible  equation  satisfied  by  Xi  and  consequently 
by  the  Xj's,  f  (x)  would  be  divisible  by  <^  (x)  and  therefore 
would  not  be  irreducible. 

By  reason  of  the  five  properties  of  ^  (x)  =  0  thus  estab- 
lished, we  may  designate  this  equation,  in  short,  as  the  irre- 
ducible equation  satisfied  by  the  x/s. 

13.     Let  us  now  compare  F  (x)  and  </>(x).    These  two  poly- 
nomials have  the  x,'s  as  their  only  roots,  and  </>  (x)  has  no 
jji     multiple  roots.      F  (x)  is,  then,  divisible  by  ^  (x)  ;  that  is, 

k  F(x)=F,(x)«^(x). 


12  FAMOUS  PROBLEMS. 

Fi  (x)  necessarily  has  rational  coefficients,  since  it  is  the  quo- 
tient obtained  by  dividing  F  (x)  by  ^  (x).  If  Fi  (x)  is  not  a 
constant  it  admits  roots  belonging  to  F  (x) ;  and  admitting 
one  it  admits  all  the  x,'s  (10).     Hence  Fi  (x)  is  also  divisible 

by  <f)  (x),  and 

Fi(x)=F2(x),/»(x). 

K  F2  (x)  is  not  a  constant  the  same  reasoning  still  holds,  the 
degree  of  the  quotient  being  lowered  by  each  operation. 
Hence  at  the  end  of  a  limited  number  of  divisions  we  reach 
an  equation  of  the  form 

F^  _  1  (x)  =  Ci  •  <^  (x), 
and  for  F  (x), 

F(x)  =  C-[,^(x)]^ 

The  polynomial  F  (x)  is  then  a  power  of  the  polynomial  of 
minimum  degree  <f>  (x),  except  for  a  constant  factor. 

14.  We  can  now  determine  the  degree  M  of  <^(x).  F  (x) 
is  of  degree  2™ ;  further,  it  is  the  vth  power  of  ^  (x).     Hence 

2'°  =  V  •  M. 

Therefore  M  is  also  a  power  of  2  and  we  obtain  the  following 
theorem  : 

The  degree  of  the  irreducible  equation  satisfied  hy  an  expres- 
sion composed  of  square  roots  only  is  always  a  power  of  2. 

15.  Since,  on  the  other  hand,  there  is  only  one  irreducible 
equation  satisfied  by  all  the  x,'s  (12,  V.),  we  have  the  converse 
theorem : 

If  an  irreducible  eqtiation  is  not  of  degree  2*,  it  cannot  be 
solved  by  square  roots. 


CHAPTER   II. 

The  Delian  Problem  and  the  Trisection  of  the  Angle. 

1.  Let  us  now  apply  the  general  theorem  of  the  preceding 
chapter  to  the  Delian  problem,  i.e.,  to  the  problem  of  the 
duplication  of  the  cube.  The  equation  of  the  problem  is 
manifestly 

3,— 

This  is  irreducible,  since  otherwise  ■v2  would  have  a 
rational  value.  For  an  equation  of  the  third  degree  which  is 
reducible  must  have  a  rational  linear  factor.  Further,  the 
degree  of  the  equation  is  not  of  the  form  2'^ ;  hence  it  cannot 
be  solved  by  means  of  square  roots,  and  the  geometric  con- 
struction with  straight  edge  and  compasses  is  impossible. 

2.  Next  let  us  consider  the  more  general  equation 

X  designating  a  parameter  which  may  be  a  complex  quantity 
of  the  form  a  +  ib.  This  equation  furnishes  us  the  analyt- 
ical expressions  for  the  geometrical  problems  of  the  multi- 
plication of  the  cube  and  the  trisection  of  an  arbitrary  angle. 
The  question  arises  whether  this  equation  is  reducible,  i.e., 
whether  one  of  its  roots  can  be  expressed  as  a  rational  func- 
tion of  X.  It  should  be  remarked  that  the  irreducibility  of 
an  expression  always  depends  upon  the  values  of  the  quan- 
tities supposed  to  be  known.  In  the  case  x^  =  2,  we  were 
dealing   with   numerical    quantities,   and   the    question   was 

3,— 

whether  v2  could  have  a  rational  numerical  value.  In  the 
equation  x'  =  X  we  ask  whether  a  root  can  be  represented  by 
a  rational  function  of  X.      In  the  first  case,  the  so-called 


14  FAMOUS  PROBLEMS. 

domain  of  rationality  comprehends  the  totality  of  rational 
numbers  ;  in  the  second,  it  is  made  up  of  the  rational  func- 
tions of  a  parameter.     If  no  limitation  is  placed  upon  this 

parameter  we  see  at  once  that  no  expression  of  the  form  -TT^y 

in  which  <^  (A.)  and  i/r(A)  are  polynomials,  can  satisfy  our 
equation.  Under  our  hypothesis  the  equation  is  therefore 
irreducible,  and  since  its  degree  is  not  of  the  form  2^,  it  can- 
not be  solved  by  square  roots. 

Let  us  now  restrict  the  variability  of  A.     Assume 
X  =  r  (cos  <^  +  i  sin  <^)  ; 


whence      Vx  =  Vr  Vcos  </>  -f  i  sin  <f>. 

Our    problem    resolves   itself    into   two,   to 

extract  the  cube  root  of  a  real  number  and 

also  that  of  a  complex  number  of  the  form 

^^^'  ^'  cos  <f>  -\-  i  sin  (f>,  both  numbers  being  regarded 

as  arbitrary.     We  shall  treat  these  separately. 

I.    The  roots  of  the  equation  x^  =  r  are 

Vr,   c  Vr,  ^  Vr, 

representing  by  c  and  t*  the  complex  cube  roots  of  unity 

-l  +  iV3      „      -l-iV3 
e= 7: ,    r= x . 


Taking  for  the  domain  of  rationality  the  totality  of  rational 
functions  of  r,  we  know  by  the  previous  reasoning  that  the 
equation  x'  =  r  is  irreducible.  Hence  the  problem  of  the 
multiplication  of  the  cube  does  not  admit,  in  general,  of  a 
construction  by  means  of  straight  edge  and  compasses, 
II.    The  roots  of  the  equation 

x^  =  cos  <^  +  i  sin  «/> 

are,  by  De  Moivre's  formula, 


THE    TRISECTION   OF   THE  ANGLE. 


15 


Xi  =  cos  -^+  I  sin-, 


Xj  =  cos 


<^+27 


+  i  sin 


<f>-\-2i 


Xg    =    COS 


«^  +  47 


I  sin 


<^  +  47 


Fig.  2. 


These  roots  are  represented  geometrically  by  the  vertices  of 
an  equilateral  triangle  inscribed  in  the  circle  with  radius 
unity  and  center  at  the  origin.    The 
figure  shows  that  to  the  root  Xi  cor- 
responds the  argument  ^.     Hence ' 
o 

the  equation 

x^  =  cos  <^  +  i  sin  <^ 

is  the  analytic  expression  of  the 
problem  of  the  trisection  of  the 
angle. 

If  this  equation  were  reducible, 
one,  at  least,  of  its  roots  could  be  represented  as  a  rational 
function  of  cos  <j>  and  sin  <^,  its  value  remaining  unchanged 
on  substituting  ^  +  27r  for  <f>.  But  if  we  effect  this  change 
by  a  continuous  variation  of  the  angle  <f},  we  see  that  the 
roots  Xi,  X2,  X3  undergo  a  cyclic  permutation.  Hence  no  root 
can  be  represented  as  a  rational  function  of  cos  ^  and  sin  ^. 
The  equation  under  consideration  is  irreducible  and  therefore 
cannot  be  solved  by  the  aid  of  a  finite  number  of  square  roots. 
Hence  the  trisection  of  the  angle  cannot  be  effected  with  straight 
edge  and  compasses. 

This  demonstration  and  the  general  theorem  evidently  hold 
good  only  when  <^  is  an  arbitrary  angle  ;  but  for  certain  spe- 
cial values  of  <^  the  construction  may  prove  to  be  possible, 


e.g.,  when  ^  =  -^- 


CHAPTER   III. 

The  Division  of  the  Circle  into  Equal  Parts. 

1.  The  problem  of  dividing  a  given  circle  into  n  equal 
parts  has  come  down  from  antiquity  ;  for  a  long  time  we 
have  known  the  possibility  of  solving  it  when  n  =  2^  3,  5,  or 
the  product  of  any  two  or  three  of  these  numbers.  In  his 
Disquisitiones  Arithmeticae,  Gauss  extended  this  series  of 
numbers  by  showing  that  the  division  is  possible  for  every 
prime  number  of  the  form  p  =  2^^  +  1  but  impossible  for  all 
other  prime  numbers  and  their  powers.  If  in  p  =  2^  +  1 
we  make  ij.:=0  and  1,  we  get  p  =  3  and  5,  cases  already 
known  to  the  ancients.  For  /t  =  2  we  get  p  =:  2^  +  1  =  17, 
a  case  completely  discussed  by  Gauss. 

For  fi  =  3  we  get  p  =  2^  +  1  =  257,  likewise  a  prime  num- 
ber. The  regular  polygon  of  257  sides  can  be  constructed. 
Similarly  for  ix  =  4:,  since  2^  +  1  ^  65537  is  a  prime  number. 
fi.  =  5,  fx  =  6,  fi^7  give  no  prime  numbers.  For  /u,  =  8  no 
one  has  found  out  whether  we  have  a  prime  number  or  not. 
The  proof  that  the  large  numbers  corresponding  to  /x  =  5,  6,  7 
are  not  prime  has  required  a  large  expenditure  of  labor  and 
ingenuity.  It  is,  therefore,  quite  possible  that  /a  =  4  is  the 
last  number  for  which  a  solution  can  be  effected. 

Upon  the  regular  polygon  of  257  sides  Richelot  published 
an  extended  investigation  in  Crelle's  Journal,  IX,  1832, 
pp.  1-26,  146-161,  209-230,  337-356.  The  title  of  the 
memoir  is  :  De  resolutione  algebraica  aequationis  x^^  =  1,  swe 
de  divisione  circuli  per  bisectionem  anguli  septies  repetitam  in 
partes  267  inter  se  aequales  commentatio  coronata. 


THE  DIVISION  OF   THE   CIRCLE.  17 

To  the  regular  polygon  of  65537  sides  Professor  Hermes 
of  Lingen  devoted  ten  years  of  his  life,  examining  with  care 
all  the  roots  furnished  by  Gauss's  method.  His  MSS.  are 
preserved  in  the  collection  of  the  mathematical  seminary  in 
Gottingen.  (Compare  a  communication  of  Professor  Hermes 
in  No.  3  of  the  Gottinger  Nachrichten  for  1894.) 

2.  We  may  restrict  the  problem  of  the  division  of  the 
circle  into  n  equal  parts  to  the  cases  where  n  is  a  prime  num- 
ber p  or  a  power  p"^  of  such  a  number.  For  if  n  is  a  com- 
posite number  and  if  /x  and  v  are  factors  of  n,  prime  to  each 
other,  we  can  always  find  integers  a  and  b,  positive  or  nega- 
tive, such  that  1  „      1   . 

whence  — = — | — . 

To  divide  the  circle  into  fxv^  n  equal  parts  it  is  sufficient  to 
know  how  to  divide  it  into  /a  and  v  equal  parts  respectively. 
Thus,  for  n  =  15,  we  have 

1  _2_3 
15  ~  3      5' 

3.  As  will  appear,  the  division  into  p  equal  parts  (p  being 
a  prime  number)  is  possible  only  when  p  is  of  the  form 
p  =  2^  +  1.  We  shall  next  show  that  a  prime  number  can 
be  of  this  form  only  when  h  =  2'*.  For  this  we  shall  make 
use  of  Fermat's  Theorem  : 

If  p  is  a  prime  number  and  a  an  integer  not  divisible  by  p, 
these  numbers  satisfy  the  congruence 

aP-i  =  -|-i  (mod.  p). 

p  —  1  is  not  necessarily  the  lowest  exponent  which,  for  a 
given  value  of  a,  satisfies  the  congruence.  If  s  is  the  lowest 
exponent  it  may  be  shown  that  s  is  a  divisor  of  p  —  1.  In 
particular,  if  s  =:  p  —  1  we  say  that  a  is  a  primitive  root  of  p, 


]Lg  FAMOUS  PROBLEMS. 

and  notice  that  for  every  prime  number  p  there  is  a  primitive 
root.     We  shall  make  use  of  this  notion  further  on. 
Suppose,  then,  p  a  prime  number  such  that 

(1)  p  =  2'>  +  l, 
and  s  the  least  integer  satisfying 

(2)  2«  =  +  1  (mod.  p). 
From  (1)  2'>  <  p  ;  from  (2)  2«  >  p. 

.-.  s>h. 
(1)  shows  that  h  is  the  least  integer  satisfying  the  congruence 

(3)  2>'  =  —  1  (mod.  p). 
From  (2)  and  (3),  by  division, 

28-h  =  _i  (niod.  p). 

.-.(4)  s-h<f.h,     s^2h. 

From  (3),  by  squaring, 

2«i  =  1  (mod.  p). 
Comparing  with  (2)  and  observing  that  s  is  the  least  expo- 
nent satisfying  congruences  of  the  form 

2^  =  1  (mod.  p), 
we  have 

(6)  s:^2h. 

.-.  s  =  2h. 

We  have  observed  that  s  is  a  divisor  of  p  — 1=2'';  the  same 
is  true  of  h,  which  is,  therefore,  a  power  of  2.  Hence  prime 
numbers  of  the  form  2^  -|- 1  are  necessarily  of  the  form 
2»^  +  l. 

4.     This  conclusion  may  be  established  otherwise.     Sup- 
pose that  h  is  divisible  by  an  odd  number,  so  that 

h  =  h'(2n  +  l); 
then,  by  reason  of  the  formula 

x2°  +  i  +  l  =  (x  +  l)  (x2°-x2"-i+.  .  .  -x  +  1), 


THE   CYCLOTOMIC  EQUATION.  19 

p  _-2h(2n  +  i)_|_  ]^  is  divisible  by  2^^' -f- 1,  and  hence  is  not  a 
prime  number. 

5.  We  now  reach  our  fundamental  proposition  : 

p  being  a  prime  nuviber,  the  division  of  the  circle  into  p  equal 
parts  hy  the  straight  edge  and  compasses  is  impossible  unless  p 
is  of  the  form 

p  =  21'  +  1  =  2=^  +  1. 

Let  us  trace  in  the  z-plane  (z  ^  x  +  iy)  a  circle  of  radius  1. 
To  divide  this  circle  into  n  equal  parts,  beginning  at  z  =  1,  is 
the  same  as  to  solve  the  equation 

This  equation  admits  the  root  z  =  1 ;  let  us  suppress  this  root 
by  dividing  by  z  —  1,  which  is  the  same  geometrically  as  to 
disregard  the  initial  point  of  the  division.  We  thus  obtain 
the  equation 

z"-i  +  z°-2+.  .  .  -|-z  +  l  =  0, 

which  may  be  called  the  cyclotomic  equation.  As  noticed 
above,  we  may  confine  our  attention  to  the  cases  where  n  is 
a  prime  number  or  a  power  of  a  prime  number.  We  shall 
first  investigate  the  case  when  n  =  p.  The  essential  point  of 
the  proof  is  to  show  that  the  above  equation  is  irreducible. 
For  since,  as  we  have  seen,  irreducible  equations  can  only  be 
solved  by  means  of  square  roots  in  finite  number  when  their 
degree  is  a  power  of  2,  a  division  into  p  parts  is  always  im- 
possible when  p  —  1  is  not  equal  to  a  power  of  2,  i.e.,  when 

p  :,&  2'^  +  1  ^  2^^  +  1. 

Thus  we  see  why  Gauss's  prime  numbers  occupy  such  an 
exceptional  position. 

6.  At  this  point  we  introduce  a  lemma  known  as  Gauss's 
Lemma.     If 

F(z)  =  z'"+  Az—'H-  Bz^-^^-.  .  .  +  Lz+M, 


20  FAMOUS  PROBLEMS. 

where  A,  B,  .  .  .  are  integers,  and  F(z)  can  be  resolved  into 
two  rational  factors  f  (z)  and  ^  (z),  so  that 

F  (z)  =  f  (z)  •<f>(z)=  (z">'  +  a^z"^-'  +  a2Z™'-2  +  .  .  .) 
(z™' +  ^iz-"- 1  + /S^z'""-^  +  .  .  .), 
then    must    the   a's    and   )8's    also    be    integers.      In   other 
words  : 

If  an  integral  expression  can  he  resolved  into  rational  factors 
these  factors  must  he  integral  expressions. 

Let  us  suppose  the  a's  and  ^'s  to  be  fractional.  In  each 
factor  reduce  all  the  coefficients  to  the  least  common  denom- 
inator. Let  Bo  and  bo  be  these  common  denominators. 
Finally  multiply  both  members  of  our  equation  by  aobo.  It 
takes  the  form 

aoboF(z)  =  fi  (z)  <^i  (z)  =  (aoz"'  +  a^z^'-^  +  .  .  .) 
(boZ™"  +  b,z™"-i +...). 

The  a's  are  integral  and  prime  to  one  another,  as  also  the  b's, 
since  ao  and  bo  are  the  least  common  denominators. 

Suppose  ao  and  bo  different  from  unity  and  let  q  be  a  prime 
divisor  of  aobo-  Further,  let  a^  be  the  first  coefficient  of  fi  (z) 
and  bfc  the  first  coefficient  of  ^i  (z)  not  divisible  by  q.  Let 
us  develop  the  product  fx  (z)  <^i  (z)  and  consider  the  coefficient 
of^m'  +  m-'-i-k      It  will  be 

ajbk-f  ai_ibk+i  +  ai_2bk+2+.-  •  +  ai+ibfe_i  +  ai+2bk_2+  •  •  • 

According  to  our  hypotheses,  all  the  terms  after  the  first  are 
divisible  by  q,  but  the  first  is  not.  Hence  this  coefficient  is  not 
divisible  by  q.  Now  the  coefficient  of  zn''+m"-i-k  jn  the  first 
member  is  divisible  by  Bobo,  i.e.,  by  q.  Hence  if  the  identity 
is  true  it  is  impossible  for  a  coefficient  not  divisible  by  q  to 
occur  in  each  polynomial.  The  coefficients  of  one  at  least  of 
the  polynomials  are  then  all  divisible  by  q.  Here  is  another 
absurdity,  since  we  have  seen  that  all  the  coefficients   are 


THE   CYCLOTOMIC  EQUATION.  21 

prime  to  one  auotlier.  Hence  we  cannot  suppose  ao  and  bo 
different  from  1,  and  consequently  the  a's  and  yS's  are  in- 
tegral. 

7.  In  order  to  show  that  the  cyclotomic  equation  is  irre- 
ducible, it  is  sufficient  to  show  by  Gauss's  Lemma  that  the 
first  member  cannot  be  resolved  into  factors  with  integral 
coefiicients.  To  this  end  we  shall  employ  the  simple  method 
due  to  Eisenstein,  in  Crelle's  Journal,  XXXIX,  p.  167,  which 
depends  upon  the  substitution 

z  =  x  +  l. 
We  obtain 

f  (z) = ^-^ = i^+ii'^ = xp- + pxp- + E^n^xp- 

Z         X  X  J.  *  ^ 


+  ...+t^£^x+p  =  0. 


All  the  coefiicients  of  the  expanded  member  except  the  first 
are  divisible  by  p  ;  the  last  coefficient  is  always  p  itself,  by 
hypothesis  a  prime  number.  An  expression  of  this  class  is 
always  irreducible. 

For  if  this  were  not  the  case  we  should  have 

f  (x  +  1)  =  (x"  +  aiX—>  +  .  •  •  +  a,„_i  X  +  aj 
(x-'  -f  bix-'-^  +  .  .  .  +  b„._i  X  +  b„.), 

where  the  a's  and  b's  are  integers. 

Since  the  term  of  zero  degree  in  the  above  expression  of 
f  (z)  is  p,  we  have  a^b^-  =  p.  p  being  prime,  one  of  the  fac- 
tors of  an,b„'  must  be  unity.     Suppose,  then. 

Equating  the  coefficients  of  the  terms  in  x,  we  have 
P(P-1)_-         b  .-^-a  b  . 


22  FAMOUS  PROBLEMS. 

The  first  member  and  the  second  term  of  the  second  being 
divisible  by  p,  a„_ib„'  must  be  so  also.  Since  bm'=±l, 
a„_i  is  divisible  by  p.  Equating  the  coefficients  of  the  terms 
in  X*  we  may  show  that  a,n_2  is  divisible  by  p.  Similarly 
we  show  that  all  of  the  remaining  coefficients  of  the  factor 
x'"  + aix'""^^  .  .  .  +  a„_i  X -f- a„  are  divisible  by  p.  But 
this  cannot  be  true  of  the  coefficient  of  x™,  which  is  1. 
The  assumed  equality  is  impossible  and  hence  the  cyclo- 
tomic  equation  is  irreducible  when  p  is  a  prime. 

8.  We  now  consider  the  case  where  n  is  a  power  of  a 
prime  number,  say  n  =  p".  We  propose  to  show  that  when 
p  >■  2  the  division  of  the  circle  into  p^  equal  parts  is  impos- 
sible. The  general  problem  will  then  be  solved,  since  the 
division  into  p*  equal  parts  evidently  includes  the  division 
into  p^  equal  parts. 

The  cyclotomic  equation  is  now 

It  admits  as  roots  extraneous  to  the  problem  those  which 
come  from  the  division  into  p  equal  parts,  i.e.,  the  roots  of 
the  equation  „      ., 

5^  =  0. 

Z  —  1 

Suppressing  these  roots  by  division  we  obtain 

zP'-l 

as  the  cyclotomic  equation.     This  may  be  written 

ZP(P-I)  _|-  zP(p-2)  _j_  _|_  2?  4-  1  =  0. 

Transforming  by  the  substitution 

z  =  x  +  l, 
we  have 

(x  -f  l)P<P-i)  +  (x  +  l)p(p-2)  4-  .  .  .  +  (x  +  l)p  -f  1  =  0. 


THE   CYCLOTOMIC  EQUATION.  23 

The  number  of  terms  being  p,  the  term  independent  of  x  after 
development  will  be  equal  to  p,  and  the  sum  will  take  the 
form 

XP(P-l)+p.;^(x), 

where  ^  (x)  is  a  polynomial  with  integral  coefficients  whose 
constant  term  is  1.  We  have  just  shown  that  such  an  expres- 
sion is  always  irreducible.  Consequently  the  new  cyclotomic 
equation  is  also  irreducible. 

The  degree  of  this  equation  is  p  (p  —  1).  On  the  other 
hand  an  irreducible  equation  is  solvable  by  square  roots  only 
when  its  degree  is  a  power  of  2.  Hence  a  circle  is  divisible 
into  p*  equal  parts  only  when  p  =  2,  p  being  assumed  to  be  a 
prime. 

The  same  is  true,  as  already  noted,  for  the  division  into  p« 
equal  parts  when  a  >  2. 


CHAPTER   IV. 

The  Construction  of  the  Regiilar  Polygon  of  17  Sides. 

1.  We  have  just  seen  that  the  division  of  the  circle  into 
equal  parts  by  the  straight  edge  and  compasses  is  possible 
only  for  the  prime  numbers  studied  by  Gauss.  It  will  now 
be  of  interest  to  learn  how  the  construction  can  actually  be 
effected. 

The  purpose  of  this  chapter,  then,  will  be  to  show  in  an 
elementary  way  how  to  inscribe  in  the  circle  the  regular  poly- 
gon of  17  sides. 

Since  we  possess  as  yet  no  method  of  construction  based 
upon  considerations  purely  geometrical,  we  must  follow  the 
path  indicated  by  our  general  discussions.  We  consider,  first 
of  all,  the  roots  of  the  cyclotomic  equation 

x»«+x»«4-.  .  .  +x2+x+l=0, 

and  construct  geometrically  the  expression,  formed  of  square 

roots,  deduced  from  it. 

We  know  that  the  roots  can  be  put  into  the  transcendental 

form 

2K7r  ,   .    .    2Kir  ,        .    „  ... 

€«  =  cos  — H-isin—  (k  =  1,  2,  ..  .16); 

and  if 

27r    ,    .     .     Iir 
<i  =  cos  —  +  I  sm  — , 

that  c^  =  c^*. 

Geometrically,  these  roots  are  represented  in  the  complex 
plane  by  the  vertices,  different  from  1,  of  the  regular  polygon 
of  17  sides  inscribed  in  a  circle  of  radius  1,  having  the  origin 


THE  REGULAR  POLYGON   OF  17  SIDES.  25 

as  center.  The  selection  of  ci  is  arbitrary,  but  for  the  con- 
struction it  is  essential  to  indicate  some  e  as  the  point  of 
departure.  Having  fixed  upon  ci,  the  angle  corresponding  to 
e^  is  K  times  the  angle  corresponding  to  e^,  which  completely 
determines  c«. 

2.  The  fundamental  idea  of  the  solution  is  the  following  : 
Forming  a  primitive  root  to  the  modulus  17  we  may  arraiige 
the  16  roots  of  the  equation  in  a  cycle  in  a  determinate  order. 

As  already  stated,  a  number  a  is  said  to  be  a  primitive  root 
to  the  modulus  17  when  the  congruence 

a«  =  +  1  (mod.  17) 

has  for  least  solution  s  =  17  —  1  =  16.  The  number  3  pos- 
sesses this  property ;  for  we  have 


(mod.  17). 


Let  us  then  arrange  the  roots  c^  so  that  their  indices  are 
the  preceding  remainders  in  order 

^3)   €9,    Cioj   Cl3)    C5,    Ci5,    Cii,    Cie,    ei4,   Cg)    €75    ^4)   ^121    ^2?    ^65    «!• 

Notice  that  if  r  is  the  remainder  of  3*  (mod.  17),  we  have 
3*  =  17q  +  r, 
whence  e,  =  ci'  =  ci' . 

If  r'  is  the  next  remainder,  we  have  similarly 

Hence  in  this  series  of  roots  each  root  is  the  cube  of  the  preceding. 

Gauss's  method  consists  in  decomposing  this  cycle  into 

sums  containing  8,  4,  2,  1  roots  respectively,  corresponding 

to  the  divisors  of  16.     Each  of  these  sums  is  called  a  period. 


3^=    3 

3^=    5 

3«  =14 

3^3=12 

32=    9 

3«=15 

3'^=    8 

3"=    2 

3^=10 

3^=11 

3"=    7 

31*=    6 

3^=13 

3«=16 

2,^''=    4 

3i«=    1 

26  FAMOUS  PROBLEMS. 

The  periods  thus  obtained  may  be  calculated  successively  as 
roots  of  certain  quadratic  equations. 

The  process  just  outlined  is  only  a  particular  case  of  that 
employed  in  the  general  case  of  the  division  into  p  equal 
parts.  The  p  —  1  roots  of  the  cyclotomic  equation  are  cyclic- 
ally arranged  by  means  of  a  primitive  root  of  p,  and  the 
periods  may  be  calculated  as  roots  of  certain  auxiliary  equa- 
tions. The  degree  of  these  last  depends  upon  the  prime  fac- 
tors of  p  —  1.  They  are  not  necessarily  equations  of  the 
second  degree. 

The  general  case  has,  of  course,  been  treated  in  detail  by 
Gauss  in  his  Disquisitiones,  and  also  by  Bachmann  in  his 
work,  Die  Lehre  von  der  Kreisteilung  (Leipzig,  1872). 

3.  In  our  case  of  the  16  roots  the  periods  may  be  formed 
in  the  following  manner :  Form  two  periods  of  8  roots  by 
taking  in  the  cycle,  first,  the  roots  of  even  order,  then  those 
of  odd  order.  Designate  these  periods  by  Xi  and  Xg,  and 
replace  each  root  by  its  index.  We  may  then  write  symbol- 
ically 

xi  =  9 +  13 +  15 +  16+   8  +  4+    2  +  1, 
X2  =  3  +  10  +    5  +  11  +  14  +  7+12  +  6. 

Operating  upon  Xi  and  Xg  in  the  same  way,  we  form  4  periods 
of  4  terms  : 

yx  =  13  +  16+    4+    1, 

y,=   9  +  15+    8+    2, 

y8  =  10  +  ll+    7+    6, 

y,=   3+    5  +  14  +  12. 

Operating  in  the  same  way  upon  the  y's,  we  obtain  8  periods 

of  2  terms  : 

zi  =  16  +  l,  Z5  =  ll+    6, 

2,  =  13  +  4,  Z6  =  10+    7, 

Z8  =  15  +  2,  Z7=   6  +  12, 

Z4=   9  +  8,  Z8=   3  +  14. 


THE  REGULAR   POLYGON  OF  17  SIDES.  27 

It  now  remains  to  show  that  these  periods  can  be  calculated 
successively  by  the  aid  of  square  roots. 

4.  It  is  readily  seen  that  the  sum  of  the  remainders  corre- 
sponding to  the  roots  forming  a  period  z  is  always  equal  to  17. 
These  roots  are  then  e,  and  €17 _r ; 

27r    ,    .     .         27r 
e,  =  COS  r  — +1  sm  r  — , 

V  =  ei7_r  =  cos  (17  —  r)  ^  +  i  sin  (17  —  r)  jy , 

27r        .     .        27r 
=  cos  r— —  I  sm  r  — . 

Hence 

€,  +  cr'  =  2  COS  r  — . 

Therefore  all  the  periods  z  are  real,  and  we  readily  obtain 
Zi  =  2cos— ,  Z5  =  2cos6— , 

z2  =  2cos4j^,  Z6  =  2cos7j^, 

zg  =  2  cos  2  j^,  Z7  =  2  cos  5  j^, 

z4  =  2cos8y^,  Z8  =  2cos3jy. 

Moreover,  by  definition, 

Xi  =  Zi  +  Zo  +  Z3  +  Z4,      X2  =^  Z5  +  Ze  +  z-  +  Zg, 

yi  =  Zl  +  Z2,    y2  =  Z3  +  Z4,    y3  =  Z5  +  Z6,    y4  =  Z7  +  Zg. 

5.  It  will  be  necessary  to  determine  the  relative  magnitude 
of  the  different  periods.  For  this  purpose  we  shall  employ 
the  following  artifice  :  We  divide  the  semicircle  of  unit  radius 
into  17  equal  parts  and  denote  by  S;,  Sg,  •  •  •  ^w  the  distances 


28 


FAMOUS  PROBLEMS. 


O  A, 


of  the  consecutive  points  of  division  Ai,  Aa,  .  •  •  An  from  the 
initial  point  of  the  semicircle,  S17  being  equal  to  the  diam- 
eter, i.e.,  equal  to  2.      The  angle 
A^AitO  has  the  same  measure  as  the 
half  of  the  arc  A«0,  which  equals 

-^ .     Hence 

S.  =  2sm-  =  2cos5-3^. 

That   this   may  be  identical  with 
2  cos  h  j^,  we  must  have 


4h 


17  —  K, 

17 -4h. 


Giving  to  h  the  values  1,  2,  3,  4,  5,  6,  7,  8,  we  find  for  k  the 
values  13,  9,  5,  1,  —  3,  —  7,  —  11,  —  15.     Hence 


Zg  =  Sg, 

24  ^  S16, 


Zg  —        07, 
Ze  ^       Sii, 

Z7  =  —  Sg, 

Zs  =  S5. 


The  figure  shows  that  S»  increases  with  the  index  ;  hence  the 
order  of  increasing  magnitude  of  the  periods  z  is 


=-4>   ^6)    ^6}    ^l>   ^2j    ^8)    •^3> 


Moreover,  the  chord  A»A»  +  p  subtends  p  divisions  of  the  semi- 
circumference  and  is  equal  to  Sp  ;  the  triangle  OA^A^  +  p  shows 
that 

and  a  fortiori 

^K  +  p  "^  ^<  +  r  ~r  iSp  +  r'" 

Calculating  the  differences  two  and  two  of  the  periods  y,  we 
easily  find 


Hence 


THE  REGULAR  POLYGON  OF  17  SIDES.  29 

yi  —  y2  =  Si3  +  Si  —  So  H-  S^  >  0, 
Yi  —  ys  =  Sis  +  Si  +  S;  +  Sii  >  0, 
Yi  —  y4  =  Si3  +  Si  +  Ss  —  Si  >  0, 
Y2  —  Ys  =  S.,  —  Si5  +  St  +  Sn  >  0, 
Y2  —  Y4  =  So  —  Si5  +  Ss  —  Ss  <  0, 
Ys  —  Y4  =  —  St  —  Sn  +  S3  —  Ss  <  0. 


Ys  <  Y2  <  Y4  <  Yi- 
Finally  we  obtain  in  a  similar  way 

X2  <  Xi. 

27r 
6.  We  now  propose  to  calculate  Zi  =  2  cos  — .  After  mak- 
ing this  calculation  and  constructing  Zi,  we  can  easily  deduce 
the  side  of  the  regular  polygon  of  17  sides.  In  order  to  find 
the  quadratic  equation  satisfied  by  the  periods,  we  proceed  to 
determine  symmetric  functions  of  the  periods. 

Associating  Zi  with   the  period  Zg  and  thus  forming  the 
period  yi,  we  have,  first, 

zi  +  Z2  =  Yi- 
Let  us  now  determine  ZiZj.  .  We  have 

ZiZ2=(16  +  l)(13  +  4), 
where  the  symbolic  product  *cp  represents 

^K    '   f  p  ^K  +  p- 

Hence  it  should  be  represented  symbolically  by  k  +  p,  remem- 
bering to  subtract  17  from  k  -j-  p  as  often  as  possible.     Thus, 

ziz,  =  12-f  3-f  14  +  5-y4. 

Therefore  Zi  and  z^  are  the  roots  of  the  quadratic  equation 

(0  z^-yiZ  +  Y4  =  0, 

whence,  since  Zi  >  Zj, 


_  Yi  +  Vyi'  ~  4Y4  _  Yi~  ^Yi'~^Y4 


20  FAMOUS  PROBLEMS. 

We  must  now  determine  yi  and  yi.  Associating  yj  with  the 
period  y^,  thus  forming  the  period  Xi,  and  yg  with  the  period 
y4,  thus  forming  the  period  Xg,  we  have,  first, 

yi  +  y2  =  xi. 
Then, 

yiy2=(13  +  16  +  4  +  l)  (9  +  15  +  8  +  2). 

Expanding  symbolically,  the  second  member  becomes  equal 
to  the  sum  of  all  the  roots  ;  that  is,  to  —  1.  Therefore  yi 
and  y2  are  the  roots  of  the  equation 

(V)  y'^-x.y-l^O, 

whence,  since  yi  >  y^, 

_xi+v^2+l  _xi-y^7+l 

yi—      2     '       y^"      2 — • 

Similarly, 

ys  +  y4  =  X2 
and 

ysy*  =  - 1. 

Hence  yg  and  yi  are  the  roots  of  the  equation 

(V)  y^-X2y-l  =  0; 

whence,  since  yt  >  yg. 


w       X2+Vx2''  +  4  X2-Vx/  +  4 

y4= — 2 — '        y«  =  — 2 — • 

It  now  remains  to  determine  Xj  and  Xj.     Since  Xi  +  Xj  is 
equal  to  the  sum  of  all  the  roots, 

Xl  +  X2  =  —  1. 

Further, 

xix,  =  (13  + 16  +  4  +  1  +  9  +  15  +8  +  2) 

(10  +  11  +  7  +  6  +  3  +  5  +  14  + 12). 
Expanding  symbolically,  each  root  occurs  4  times,  and  thus 
XjXa  =  —  4. 


THE  REGULAR  POLYGON  OF  17  SIDES.  31 

Therefore  Xx  aud  X2  are  the  roots  of  the  quadratic 

(i)  x=^+x-4  =  0; 

whence,  since  x^  >  Xj, 

_-l  +  Vl7  _  -  1  -  Vl7 


Solving  equations  i,  rj,  rj',  ^  in  succession,  Zi  is  determined 
by  a  series  of  square  roots. 

Effecting  the  calculations,  we  see  that  Zi  depends  upon  the 
four  square  roots 


Vl7,   Vxi2  +  4,   Vx7+4,   Vyi^-4y4. 

If  we  wish  to  reduce  Zj  to  the  normal  form  we  must  see 
whether  any  one  of  these  square  roots  can  be  expressed 
rationally  in  terms  of  the  others. 

Now,  from  the  roots  of  (17), 


Vxi^  -I-  4  =  y^  —  yjj, 


Vxa''  +  4  =  yi  —  ys. 
Expanding  symbolically,  we  verify  that 

(yi  -  ya)  (y4  —  ys)  =  2  (xi  -  x^),* 

*  (yi  -  y2)  (y4  -  ys)  =  (13  +  16  +  4  +  1  -  9  -  15  -  8  -  2)  (3  +  5  +  14 
+  12-10-11-7-6) 

=  16+  1  +  10+  8-  6-  7-  3-  2 
+  2+  4+13+11-  9-10-  6-  5 
+  7  +  9  +  1  +  16-14-15-11-10 
+  4+  6+15  +  13-11-12-  8-  7 
-12-14-    6-    4+    2+    3+16  +  15 

-  1-    3-12-10+    8+    9+    5+    4 

-  11  -  13  -    5  -    3  +    1  +    2  +  15+14 

-  5  -    7  -  16  -  14  +  12  +  13  +    9  +    8 

=  2(16  +  1  +  8  +  2  +  4+  13  +  15+9-10-6-7-3-11-5-14 

-  12) 
=  2(xi  -  X2). 


w  FAMOUS  PROBLEMS. 

that  is,  

Vxi=^  +  4  Vx2*  +  4  =  2Vl7. 

Hence  Vxj"  +  4  can  be  expressed  rationally  in  terms  of  the 
other  two  square  roots.  This  equation  shows  that  if  two  of 
the  three  differences  y^  —  y^,  y*  —  ys,  ^i  —  ^2  are  positive,  the 
same  is  true  of  the  third,  which  agrees  with  the  results  ob- 
tained directly. 

Replacing  now  xi,  yi,  yt  by  their  numerical  values,  we 
obtain  in  succession 

-I  +  V17 
xi—  2  ' 

-  1  -f  Vl7  +  \/34  -  2  Vrf 

yi-  4 

-1-V17  + V34  +  2VI7 

- 1  +  Vrf  +  V34  -  2  Viz 
z,=— f^ 

JeS  fl2Vl7— I6V34+2V17— 2(1— Vl7)V34  — 2VT7 

The  algebraic  part  of  the  solution  of  our  problem  is  now 
completed.  We  have  already  remarked  that  there  is  no  known 
construction  of  the  regular  polygon  of  17  sides  based  upon 
purely  geometric  considerations.  There  remains,  then,  only 
the  geometric  translation  of  the  individual  algebraic  steps. 

7.  We  may  be  allowed  to  introduce  here  a  brief  historical 
account  of  geometric  constructions  with  straight  edge  and 
compasses. 

In  the  geometry  of  the  ancients  the  straight  edge  and  com- 
passes were  always  used  together ;  the  difficulty  lay  merely  in 
bringing  together  the  different  parts  of  the  figure  so  as  not  to 


THE  REGULAR  POLYGON  OF  17  SIDES.  33 

draw  any  unnecessary  lines.  Whether  the  several  steps  in 
the  construction  were  made  with  straight  edge  or  with  com- 
passes was  a  matter  of  indifference. 

On  the  contrary,  in  1797,  the  Italian  Mascheroni  succeeded 
in  effecting  all  these  constructions  with  the  compasses  alone  ; 
he  set  forth  his  methods  in  his  Geometria  del  compasso,  and 
claimed  that  constructions  with  compasses  were  practically 
more  exact  than  those  with  the  straight  edge.  As  he  ex- 
pressly stated,  he  wrote  for  mechanics,  and  therefore  with  a 
practical  end  in  view.  Mascheroni's  original  work  is  difficult 
to  read,  and  we  are  under  obligations  to  Hutt  for  furnishing 
a  brief  resume  in  German,  Die  MascheronV schen  Constructionen 
(Halle,  1880). 

Soon  after,  the  French,  especially  the  disciples  of  Carnot, 
the  author  of  the  Geometrie  de  position,  strove,  on  the  other 
hand,  to  effect  their  constructions  as  far  as  possible  with 
the  straight  edge.  (See  also  Lambert,  Freie  Perspective, 
1774.) 

Here  we  may  ask  a  question  which  algebra  enables  us  to 
answer  immediately  :  In  what  cases  can  the  solution  of  an 
algebraic  problem  be  constructed  with  the  straight  edge  alone  ? 
The  answer  is  not  given  with  sufficient  explicitness  by  the 
authors  mentioned.     We  shall  say  : 

With  the  straight  edge  alone  we  can  construct  all  algebraic 
expressions  whose  form  is  rational.  — - 

With  a  similar  view  Brianchon  published  in  1818  a  paper, 
Les  applicatio7is  de  la  theorie  des  transversales,  in  which  he  shows 
how  his  constructions  can  be  effected  in  many  cases  with  the 
straight  edge  alone.  He  likewise  insists  upon  the  practical 
value  of  his  methods,  which  are  especially  adapted  to  field 
work  in  surveying. 

Poncelet  was  the  first,  in  his  Traite  des  proprietes  projectives 
(Vol.  I,  Kos.  351-357),  to  conceive  the  idea  that  it  is  sufficient 
to  use  a  single  fixed  circle  in  connection  with  the  straight  lines 


34  FAMOUS  PROBLEMS. 

of  the  plane  in  order  to  construct  all  expressions  depending 
upon  square  roots,  the  center  of  the  fixed  circle  being  given. 

This  thought  was  developed  by  Steiner  in  1833  in  a  cele- 
brated paper  entitled  Die  geometrischen  Constructionen,  ausge- 
fuJiH  mittels  der  geraden  Linie  und  eines  festen  Kreises,  als 
Lehrgegenstand  fur  hohere  Unterrichtsanstalten  und  zum  Selbst- 
unterricht. 

8.  To  construct  the  regular  polygon  of  17  sides  we  shall 
follow  the  method  indicated  by  von  Staudt  (Crelle's  Journal, 
Vol.  XXIV,  1842),  modified  later  by  Schroter  (Crelle's  Jottr- 
nal,  Vol.  LXXV,  1872).  The  construction  of  the  regular 
polygon  of  17  sides  is  made  in  accordance  with  the  methods 
indicated  by  Poncelet  and  Steiner,  inasmuch  as  besides  the 
straight  edge  but  one  fixed  circle  is  used.* 

First,  we  will  show  how  with  the  straight  edge  and  one  fixed 
circle  we  can  solve  every  quadratic  equation. 

At  the  extremities  of  a  diameter  of  the  fixed  unit  circle 
(Fig.  4)  we  draw  two  tangents,  and  select  the  lower  as  the 

axis  of  X,  and  the  diameter 
perpendicular  to  it  as  the 
axis  of  Y.  Then  the  equa- 
tion of  the  circle  is 

x«-fy(y-2)=0. 
Let 

x^  —  px  +  q  =  0 

be  any  quadratic  equation 
with  real  roots  Xj  and  Xa.  Required  to  construct  the  roots  Xi 
and  Xg  upon  the  axis  of  X. 

Lay  off  upon  the  upper  tangent  from  A  to  the  right,  a  seg- 

4 
ment  measured  by  -  ;  upon  the  axis  of  X  from  0,  a  segment 

*  A  Mascheroni  construction  of  the  regular  polygon  of  17  sides  by 
L.  Gerard  is  given  in  Math.  Annalen,  Vol.  XL VIII,  1896,  pp.  390-392. 


THE  REGULAR  POLYGON   OF  17  SIDES.  35 

measured  by  —  ;  connect  the  extremities  of  these  segments  by 

the  line  3  and  project  the  intersections  of  this  line  with  the 
circle  from  A,  by  the  lines  1  and  2,  upon  the  axis  of  X.  The 
segments  thus  cut  off  upon  the  axis  of  X  are  measured  by  Xi 
and  Xj. 

Proof.     Calling  the  intercepts  upon  the  axis  of  X,  Xi  and  X2, 
we  have  the  equation  of  the  line  1, 

2x  +  Xi(y-2)=0; 
of  the  line  2, 

2x  +  x2(y-2)  =  0. 

If  we  multiply  the  first  members  of  these  two  equations  we 
get 

x^+^^^x(y-2)H-'^(y-2)^  =  0 

as  the  equation  of  the  line  pair  formed  by  1  and  2.  Subtract- 
ing from  this  the  equation  of  the  circle,  we  obtain 

^^^x(y-2)  +  ^(y-2)^-y(y-2)  =  0. 

This  is  the  equation  of  a  conic  passing  through  the  four 
intersections  of  the  lines  1  and  2  with  the  circle.  From 
this  equation  we  can  remove  the  factor  y  —  2,  correspond- 
ing to  the  tangent,  and  we  have  left 

^^x-f'^^(y-2)-y  =  0, 

which  is  the  equation  of  the  line  3.  If  we  now  make 
"^i  4"  Xa  =  p  and  X1X2  =  q,  we  get 

^x  +  ^(y-2)-y  =  0, 
and  the  transversal  3  cuts  off  from  the  line  y  :=  2  the  seg- 


se 


FAMOUS  PROBLEMS. 


ment  -  ,  and  from  the  line  y  =  0  the  segment  3.     Thus  the 

P  .  .  ^ 

correctness  of  the  construction  is  established. 

9.  In  accordance  with  the  method  just  explained,  we  shall 
now  construct  the  roots  of  our  four  quadratic  equations. 
They  are  (see  pp.  29-31) 

(f)  x^  +  X  —  4  =  0,  with  roots  xi  and  Xg ;  Xi  >  Xg, 
iv)  y^  —  Xiy  —  1  =0,  with  roots  yi  and  ya ;  yi  >  yi, 
Iv')  y^  —  X2y  —  1  =  0,  with  roots  yg  and  yt',  y*  >  ya, 
(^)  z^  —  yiz  +  y4  =  0,  with  roots  Zi  and  Za ;  Zi  >  Zj. 
These  will  furnish 

zi  =  2  cos  — , 

whence  it  is  easy  to  construct  the  polygon  desired.  We 
notice  further  that  to  construct  Zj_  it  is  sufficient  to  construct 
Xi>  Xj,  yi,  y4. 

We  then  lay  off  the  following  segments  :  upon  the  upper 
tangent,  y  =  2, 

_       £   £    £ 

Xi    Xj    yi 
upon  the  axis  of  X, 

+  4    _i    _i    U 

^    '        X.'        X,'  y,- 

This  may  all  be  done  in  the  following  manner :  The 
straight  line  connecting  the  point  -|-  4  upon  the  axis  of  X 
with  the  point  —  4  upon  the  tangent  y  =  2  cuts  the  circle  in 


THE  REGULAR  POLYGON   OF  17   SIDES.  37 

two  points,  the  projection  of  which  from  the  point  A  (0,  2), 
the  upper  vertex  of  the  circle,  gives  the  two  roots  Xi,  Xj  of  the 
first  quadratic  equation  as  intercepts  upon  the  axis  of  X. 

4 

To  solve  the  second  equation  we  have  to  lay  off  —  above 

and below. 

To  determine  the  first  point  we  connect  Xi  upon  the  axis  of 
X  with  A,  the  upper  vertex,  and  from  0,  the  lower  vertex, 
draw  another  straight  line  through  the  intersection  of  this 
line  with  the  circle.     This  cuts  off  upon  the  upper  tangent 

4 

the  intercept  — .     This  can  easily  be  shown  analytically. 

Xi 

The  equation  of  the  line  from  A  to  Xj  (Fig.  5), 
2x  +  xiy=r2xi, 
and  that  of  the  circle, 

x2  +  y(y-2)=.0, 
give  as  the  coordinates  of  their  intersection 

4xi  2xi^ 

Xi'  +  4'  Xx^  +  4" 

The  equation  of  the  line  from  0  through  this  point  becomes 


Xl 


2 

4 


cutting  off  upon  y  =  2  the  intercept 


We  reach  the  same  conclusion  still  more  simply  by  the  use 
of  some  elementary  notions  of  projective  geometry.  By  our 
construction  we  have  obviously  associated  with  every  point  x 
of  the  lower  range  one,  and  only  one,  point  of  the  upper,  so 
that  to  the  point  x  =:  oo  corresponds  the  point  x'  =  0,  and  con- 
versely.    Since  in  such  a  correspondence  there  must  exist  a 


38  FAMOUS  PROBLEMS. 

linear  relation,  the  abscissa  x'  of  the  upper  point  must  satisfy 

the  equation                              const. 
x'  = • 

X 

.    Since  x'  =  2  when  x  =  2,  as  is  obvious  from  the  figure,  the 
constant  =  4. 


Fig.  6. 


1 


To  determine upon  the  axis  of  X  we  connect  the  point 

—  4  upon  the  upper  with  the  point  + 1  upon  the  lower  tan- 
gent (Fig.  6).     The  point  thus  determined  upon  the  vertical 

4 
diameter  we  connect  with  the   point  —  above.     This   line 

cuts  off  upon  the  axis  of   X  the  intercept .      For   the 

line  from  —  4  to  +  1, 

5y  +  2x  =  2, 

intersects  the  vertical  diameter  in  the  point  (0,  f).     Hence 
the  equation  of  the  line  from  —  to  this  point  is 

Xi 

5y  — 2xix  =  2, 
and  its  intersection  with  the  lower  tangent  gives . 

Xi 

The  projection  from  A  of  the  intersections  of  the  line  from 

1        4      . 
~  —  to  —  with  the  circle  determines  upon  the  axis  of  X  the 

two  roots  of  the  second  quadratic  equation,  of    which,  as 


TEE  REGULAR  POLYGON  OF  17  SIDES. 


39 


already  noted,  we  need  only  the  greater,  y^.  This  corres- 
ponds, as  shown  by  the  figure,  to  the  projection  of  the  upper 
intersection  of  our  transversal  with  the  circle. 

Similarly,  we  obtain  the  roots  of  the  third  quadratic  equa- 
tion. Upon  the  upper  tangent  we  project  from  0  the  inter- 
section of  the  circle  with  the  straight  line  which  gave  upon 
the  axis  of  X  the  root  +  Xg.     This  immediately  gives  the 

4 
intercept  — ,  by  reason  of  the  correspondence  just  explained. 

X2 


Fig.  7. 

If  we  connect  this  point  with  the  point  where  the  vertical 
diameter  intersects  the  line  joining  —  4  above  and  +  1  below, 

we  cut  off  upon  the  axis  of  X  the  segment ,  as  desired. 

X2 

If  we  project  that  intersection  of  this  transversal  with  the 

circle  which  lies  in  the  positive  quadrant  from  A  upon  the 

axis  of  X,  we  have  constructed  the  required  root  yt  of  the  third 

quadratic  equation. 

We  have  finally  to  determine  the  root  Zi  of  the  fourth  quad- 

4  y, 

ratio  equation  and  for  this  purpose  to  lay  off  —  above  and  — 

below.  We  solve  the  first  problem  in  the  usual  way,  by  pro- 
jecting the  intersection  of  the  circle  with  the  line  connecting 
A  with  -\-  yi  below,  from  0  upon  the  upper  tangent,  thus 

4 
obtaining  — .     For  the  other  segment  we  connect  the  point 

-\-  4  above  with  yt  below,  and  then  the  point  thus  determined 


40 


FAMOUS  PROBLEMS. 


upon  the  vertical  diameter  produced  with  — .     This  line  cuts 

off  upon  the  axis  of  X  exactly  the  segment  desired,  — .     For 
the  line  a  (Fig.  8)  has  the  equation  ^ 

(y4-4)y  +  2x  =  2y4. 


Fig.  8. 


2y, 


y* 


It  cuts  off  upon  the  vertical  diameter  the  segment 
The  equation  of  the  line  b  is  then 

2yix  +  (y4-4)y  =  2y4, 
and  its  intersection  with  the  axis  of  X  has  the  abscissa  — . 

yi 

If  we  project  the  upper  intersection  of  the  line  b  with  the 

27r 

circle  from  A  upon  the  axis  of  X,  we  obtain  Zj  =  2  cos  — . 

If  we  desire  the  simple  cosine  itself  we  have  only  to  draw  a 
diameter  parallel  to  the  axis  of  X,  on  which  our  last  projecting 


A  perpendicular  erected  at  this 


o 

ray  cuts  off  directly  cos  — . 

point  gives  immediately  the  first  and  sixteenth  vertices  of  the 
regular  polygon  of  17  sides. 

The  period  Zi  was  chosen  arbitrarily  ;  we  might  construct 
in  the  same  way  every  other  period  of  two  terms  and  so  find 
the  remaining  cosines.  These  constructions,  made  on  separate 
figures  so  as  to  be  followed  more  easily,  have  been  combined 
in  a  single  figure  (Fig.  9),  which  gives  the  complete  construc- 
tion of  the  regular  polygon  of  17  sides. 


THE  REGULAR  POLYGON  OF  17  SIDES. 


41 


CHAPTER   V. 

General  Considerations  on  Algebraic  Constructions. 

1 .  We  shall  now  lay  aside  the  matter  of  construction  with 
straight  edge  and  compasses.  Before  quitting  the  subject  we 
may  mention  a  new  and  very  simple  method  of  effecting  cer- 
tain constructions,  'paper  folding.  Hermann  AYiener  has 
shown  how  by  paper  folding  we  may  obtain  the  network  of 
the  regular  polyhedra.  Singularly,  about  the  same  time  a 
Hindu  mathematician,  Sundara  Row,  of  Madras,  published  a 
little  book.  Geometrical  Exercises  in  Paper  Eoldinff*  (MadrsLS, 
Addison  &  Co.,  1893),  in  which  the  same  idea  is  consider- 
ably developed.  The  author  shows  how  by  paper  folding  we 
may  construct  by  points  such  curves  as  the  ellipse,  cissoid,  etc. 

2.  Let  us  now  inquire  how  to  solve  geometrically  prob- 
lems whose  analytic  form  is  an  equation  of  the  third  or  of 
higher  degree,  and  in  particular,  let  us  see  how  the  ancients 
succeeded.  The  most  natural  method  is  by  means  of  the 
conies,  of  which  the  ancients  made  much  use.  For  example, 
they  found  that  by  means  of  these  curves  they  were  enabled 
to  solve  the  problems  of  the  duplication  of  the  cube  and  the 
trisection  of  the  angle.  "We  shall  in  this  place  give  only  a 
general  sketch  of  the  process,  making  use  of  the  language 
of  modern  mathematics  for  greater  simplicity. 

Let  it  be  required,  for  instance,  to  solve  grapliically  the 

cubic  equation  .  . 

x«  +  ax'' +  bx  +  c  =  0, 

or  the  biquadratic, 

x*+ax«+bx2  +  cx  +  d  =  0. 

*  See  American  edition,  revised  by  Beman  and  Smith,  The  Open  Court 
Publishing  Co.,  Chicago,  1901. 


ALGEBRAIC   CONSTRUCTIONS. 
Fut  X*  =  y  ;  our  equations  become 


43 


and 


xy  -|-  ay  +  bx  +  c  =  0 
y^  +  axy  +  by  +  ex  -f  d  =  0. 


The  roots  of  the  equations  proposed  are  thus  the  abscissas 
of  the  points  of  intersection  of  the  two  conies. 
The  equation 


represents  a  parabola  with  axis  vertical.     The  second  equa- 
tion, 

xy  +  ay  +  bx  +  c  =  0, 

represents  an  hyperbola  whose  asymptotes  are  parallel  to  the 
axes  of  reference  (Fig.  10).     One  of  the  four  points  of  inter- 


FiG.  10. 


section  is  at  infinity  upon  the  axis  of  Y,  the  other  three  at  a 
finite  distance,  and  their  abscissas  are  the  roots  of  the  equa- 
tion of  the  third  degree. 

In  the  second  case  the  parabola  is  the  same.  The  hyper- 
bola (Fig.  11)  has  again  one  asymptote  parallel  to  the  axis  of 
X  while  the  other  is  no  longer  perpendicular  to  this  axis. 
The  curves  now  have  four  points  of  intersection  at  a  finite 
distance. 


44 


FAMOUS  PROBLEMS. 


The  methods  of  the  ancient  mathematicians  are  given  in 
detail  in  the  elaborate  work  of  M.  Cantor,  Geschichte  der 
Mathematik  (Leipzig,  1894,  2d  ed.).  Especially  interesting  is 
Zeuthen,  Die  Kegelschnitte  im  Altertum  (Kopenhagen,  1886, 
in  German  edition).  As  a  general  compendium  we  may  men- 
tion Baltzer,  Analytische  Geometrie  (Leipzig,  1882). 


Beside  the  conies,  the  ancients  used  for  the  solution  of 
the  above-mentioned  problems,  higher 
curves  constructed  for  this  very  pur- 
pose. We  shall  mention  here  only 
the  Cissoid  and  the  Conchoid. 

The  cissoid  of  Diodes  (c.  150  b.c.) 
may  be  constructed  as  follows  (Fig. 
12)  :  To  a  circle  draw  a  tangent  (in  the 
figure  the  vertical  tangent  on  the  right) 
and  the  diameter  perpendicular  to  it. 
Draw  lines  from  0,  the  vertex  of  the 
circle  thus  determined,  to  points  upon 
the  tangent,  and  lay  off  from  0  upon 
each  the  segment  lying  between  its 
intersection  with  the  circle  and  the 
tangent.  The  locus  of  points  so  deter- 
mined is  the  cissoid. 

To  derive  the  equation,  let  r  be  the 
radius  vector,  6  the  angle  it  makes  with 
the  axis  of  X.  If  we  produce  r  to  the 
tangent  on  the  right,  and  call  the  diameter  of  the  circle  1, 

1 


Fig.  12. 


the  total  segment  equals 


circle  is  cos  $. 
hence 


The  portion  cut  off  by  the 
r,  and 


cos  0 
The  difference  of  the  two  segments  is 


cos  6 


~-  cos  d  ■■ 


sin'g 
cos  $' 


ALGEBRAIC   CONSTRUCTIONS.  45 

By  transformation  of  coordinates  we  obtain  the  Cartesian 
equation, 

(x2-f-y2)x-y2  =  0. 

The  curve  is  of  the  third  order,  lias  a  cusp  at  the  origin, 
and  is  symmetric  to  the  axis  of  X.  The  vertical  tangent  to 
the  circle  with  which  we  began  our  construction  is  an  asymp- 
tote. Finally  the  cissoid  cuts  the  line  at  infinity  in  the  cir- 
cular points. 

To  show  how  to  solve  the  Delian  problem  by  the  use  of 
this  curve,  we  write  its  equation  in  the  following  form  : 


0) 


X  j  1  —  X 


We  now  construct  the  straight  line, 


^-  =  X. 


This  cuts  off  upon  the  tangent  x  =  1  the  segment  X,  and 
intersects  the  cissoid  in  a  point  for  which 


A^ 


This  is  the  equation  of  a  straight  line  passing  through  the 
point  y  =  0,  x  :=  1,  and  hence  of  the  line  joining  this  point 
to  the  point  of  the  cissoid. 

This  line  cuts  off  upon  the  axis  of  Y  the  intercept  X^ 
We  now  see  how  f^  may  be  constructed.  Lay  off  upon 
the  axis  of  Y  the  intercept  2,  join  this  point  to  the  point 
x  =  1,  y  =  0,  and  through  its  intersection  with  the  cissoid 
draw  a  line  from  the  origin  to  the  tangent  x  =  1.  The  inter- 
cept on  this  tangent  equals  1^. 

4.     The  conchoid  of  Nicomedes  (c.  150  B.C.)  is  constructed 
as  follows  :  Let  0  be  a  fixed  point,  a  its  distance  from  a  fixed 


46 


FAMOUS  PROBLEMS. 


line.  If  we  pass  a  pencil  of  rays  through  O  and  lay  off  on 
each  ray  from  its  intersection  with  the  fixed  line  in  both 
directions  a  segment  b,  the  locus  of  the  points  so  determined 
is  the  conchoid.     According  as  b  is  greater  or  less  than  a, 

the  origin  is  a  node  or  a  con- 
jugate point ;  for  b  =  a  it  is 
a  cusp  (Fig.  13). 

Taking  for  axes  of  X  and  Y 
the  perpendicular  and  paral- 
lel through  0  to  the  fix-»4 
line,  we  have 


whence 

(x='  +  y')(x-a)=^-bV  =  0. 

The  conchoid  is  then  of  the 
fourth  order,  has  a  double 
point  at  the  origin,  and  is 
composed  of  two  branches 
having  for  common  asymptote 
the  line  x  =  a.  Further,  the 
factor  (x^  +  y'O  shows  that  the 
curve  passes  through  the  cir- 
cular points  at  infinity,  a  mat- 
ter of  immediate  importance. 
We  may  trisect  any  angle  by  means  of  this  curve  in  the 
following  manner  :  Let  <^=  MOY  (Fig.  13)  be  the  angle  to 
be  divided  into  three  equal  parts.  On  the  side  OM  lay  off 
OM  =  b,  an  arbitrary  length.  With  M  as  a  center  and  radius 
b  describe  a  circle,  and  through  M  perpendicular  to  the  axis 
of  X  with  origin  0  draw  a  vertical  line  representing  the 
asymptote  of  the  conchoid  to  be  constructed.     Construct  the 


Fig.  13. 


ALGEBRAIC   CONSTRUCTIONS.  47 

conchoid.  Connect  0  with  A,  the  intersection  of  the  circle 
and  the  conchoid.  Then  is  Z.  AOY  one  third  of  Z.  <t>,  as  is 
easily  seen  from  the  figure. 

Our  previous  investigations  have  shown  us  that  the  prob- 
lem of  the  trisection  of  the  angle  is  a  problem  of  the  third 
degree.     It  admits  the  three  solutions 

<f>      4>  +  27r      <f>-\-4:7r 

3'   ^r~'  "^ 

Every  algebraic  construction  which  solves  this  problem  by 
the  aid  of  a  curve  of  higher  degree  must  obviously  furnish  all 
the  solutions.  Otherwise  the  equation  of  the  problem  would 
not  be  irreducible.  These  different  solutions  are  shown  in 
the  figure.  The  circle  and  the  conchoid  intersect  in  eight 
points.  Two  of  them  coincide  with  the  origin,  two  others 
with  the  circular  points  at  infinity.  None  of  these  can  give 
a  solution  of  the  problem.  There  remain,  then,  four  points 
of  intersection,  so  that  we  seem  to  have  one  too  many.  This 
is  due  to  the  fact  that  among  the  four  points  we  necessarily 
find  the  point  B  such  that  0  M  B  =  2  b,  a  point  which  may  be 
determined  without  the  aid  of  the  curve.  There  actually 
remain  then  only  three  points  corresponding  to  the  three 
roots  furnished  by  the  algebraic  solution. 

5.  In  all  these  constructions  with  the  aid  of  higher  alge- 
braic curves,  we  must  consider  the  practical  execution.  We 
need  an  instrument  which  shall  trace  the  curve  by  a  con- 
tinuous movement,  for  a  construction  by  points  is  simply  a 
method  of  approximation.  Several  instruments  of  this  sort 
have  been  constructed ;  some  were  known  to  the  ancients. 
Nicomedes  invented  a  simple  device  for  tracing  the  conchoid. 
It  is  the  oldest  of  the  kind  besides  the  straight  edge  and 
compasses.  (Cantor,  I,  p.  302.)  A  list  of  instruments  of 
more  recent  construction  may  be  found  in  Dyck's  Katalog, 
pp.  227-230,  340,  and  Nachtrag,  pp.  42,  43. 


PART   II. 

TBANSCENDENTAL  NUMBEBS  AND  THE  QUADBATTTBE  OF  THE 

CIBCLE. 


CHAPTER   I. 

Cantor's   Demonstration   of  the   Existence   of 
Transcendental  Numbers. 

1.  Let  us  represent  numbers  as  usual  by  points  upon  the 
axis  of  abscissas.  If  we  restrict  ourselves  to  rational  numbers 
the  corresponding  points  will  fill  the  axis  of  abscissas  densely 
throughout  (uberall  dicht),  i.e.,  in  any  interval  no  matter  how 
small  there  is  an  infinite  number  of  such  points.  Neverthe- 
less, as  the  ancients  had  already  discovered,  the  continuum 
of  points  upon  the  axis  is  not  exhausted  in  this  way  ;  between 
the  rational  numbers  come  in  the  irrational  numbers,  and  the 
question  arises  whether  there  are  not  distinctions  to  be  made 
among  the  irrational  numbers. 

Let  us  define  first  what  we  mean  by  algebraic  numbers. 
Every  root  of  an  algebraic  equation 

aoto"  +  aia>"-i  +  •  •  •  +  a„_i(o  +  a„  =  0 

with  integral  coefficients  is  called  an  algebraic  number.  Of 
course  we  consider  only  the  real  roots.  Rational  numbers 
occur  as  a  special  case  in  equations  of  the  form 

BqW  -|-  21;  =  0. 


go  FAMOUS  PROBLEMS. 

We  now  ask  the  question :  Does  the  totality  of  real 
algebraic  numbers  form  a  continuum,  or  a  discrete  series 
such  that  other  numbers  may  be  inserted  in  the  intervals? 
These  new  numbers,  the  so-called  transcendental  numbers, 
would  then  be  characterized  by  this  property,  that  they  cannot 
be  roots  of  an  algebraic  equation  with  integral  coefficients. 

This  question  was  answered  first  by  Liouville  (Comptes 
rendus,  1844,  and  Liouville's  Journal,  Vol.  XVI,  1851),  and 
in  fact  the  existence  of  transcendental  numbers  was  demon- 
strated by  him.  But  his  demonstration,  which  rests  upon  the 
theory  of  continued  fractions,  is  rather  complicated.  The 
investigation  is  notably  simplified  by  using  the  developments 
given  by  Georg  Cantor  in  a  memoir  of  fundamental  impor- 
tance, Ueber  eine  Eigenschaft  des  Inbegnffes  reeller  algebra- 
ischer  Zahlen  (Crelle's  Journal,  Vol.  LXXVII,  1873).  We 
shall  give  his  demonstration,  making  use  of  a  more  simple 
notion  which  Cantor,  under  a  different  form,  it  is  true,  sug- 
gested at  the  meeting  of  naturalists  in  Halle,  1891. 

2.  The  demonstration  rests  upon  the  fact  that  algebraic 
numbers  form  a  countable  mass,  while  transcendental  numbers 
do  not.  By  this  Cantor  means  that  the  former  can  be  arranged 
in  a  certain  order  so  that  each  of  them  occupies  a  definite 
place,  is  numbered,  so  to  speak.  This  proposition  may  be 
stated  as  follows : 

The  manifoldness  of  real  algebraic  numbers  and  the  mani- 
foldness  of  positive  integers  can  be  brought  into  a  one-to-one 
correspondence. 

We  seem  here  to  meet  a  contradiction.  The  positive  inte- 
gers form  only  a  portion  of  the  algebraic  numbers ;  since 
each  number  of  the  first  can  be  associated  with  one  and  one 
only  of  the  second,  the  part  would  be  equal  to  the  whole. 
This  objection  rests  upon  a  false  analogy.  The  proposition 
that  the  part  is  always  less  than  the  whole  is  not  true  for 


TRANSCENDENTAL  NUMBERS.  51 

infinite  masses.  It  is  evident,  for  example,  that  we  may 
establish  a  one-to-one  correspondence  between  the  aggregate 
of  positive  integers  and  the  aggregate  of  positive  even  num- 
bers, thus : 

0     1     2     3  •  •  •      n  •  •  • 

0     2     4     6  •  •  •  2n  •  •  • 

In  dealing  with  infinite  masses,  the  words  (/reat  and  small  are 
inappropriate.  As  a  substitute.  Cantor  has  introduced  the 
word  power  (Mdchtigkeit),  and  says  :  Two  infinite  masses  have 
the  same  poiver  when  they  can  be  brought  into  a  one-to-one  cor- 
respondence  with  each  other.  The  theorem  which  we  have  to 
prove  then  takes  the  following  form  :  Ths  aggregrate  of  real 
algebraic  numbers  has  the  same  power  as  the  aggregate  of 
positive  integers. 

We  obtain  the  aggregate  of  real  algebraic  numbers  by  seek- 
ing the  real  roots  of  all  algebraic  equations  of  the  form 

aow"  +  ajo)"-^  +  •  •  •  +  a„_ia)  -f-  a„  =  0  ; 

all  the  a's  are  supposed  prime  to  one  another,  ao  positive, 
and  the  equation  irreducible.  To  arrange  the  numbers  thus 
obtained  in  a  definite  order,  we  consider  their  height  N  as 
defined  by 

N  =  n-l+|aoi  +  lai|  +  .  •  -H-la^I, 

laj  representing  the  absolute  value  of  a^,  as  usual.  To  a 
given  number  N  corresponds  a  finite  number  of  algebraic 
equations.  For,  N  being  given,  the  number  n  has  certainly 
an  upper  limit,  since  N  is  equal  to  n  —  1  increased  by  positive 
numbers ;  moreover,  the  difference  N  —  (n  —  1)  is  a  sum  of 
positive  numbers  prime  to  one  another,  whose  number  is 
obviously  finite. 


52 


FAMOUS  PROBLEMS. 


N 

n 

laol 

lail 

|a,l 

|a,| 

'a«i 

Equation. 

*(N) 

Roots. 

1 

1 

1 

0 

x  =  0 

1 

0 

2 

0 

0 

0 

— 

2 

1 

2 

1 

0 

1 

x±l=0 

2 

—  1 

+  1 

2 

1 

0 

0 

— 

3 

1 

3 
2 

1 

0 

1 

2 

2a;±l=0 
x±2=0 

4 

—  2 

1 
2 

+  1 

2 

2 

1 
1 

0 

1 
0 

0 
0 

1 

- 

+  2 

3 

1 

0 

0 

0 

— 

4 

1 

4 
3 
2 

1 

0 

1 

2 
3 

3z±l  =0 
x±3  =  0 

12 

-8 

-  1.61803 

-  1.41421 

-  0.70711 

2 

3 

0 

0 

— 

-  0.61803 

2 

1 

0 

— 

-  0.33333 

2 

0 

1 

2x2-1=0 

+  0.33333 

1 

2 

0 

— 

+  0.61803 

1 

1 

1 

x2  ±  a;  -  1  =  0 

+  0.70711 

1 

0 

2 

x2  -  2  =  0 

+  1.41421 

3 

2 

0 

0 

0 

— 

+  1.61803 

1 

1 

0 

0 

— 

+  3 

1 

0 

1 

0 

— 

1 

0 

0 

1 

— 

4 

1 

0 

0 

0 

0 

— 

Among  these  equations  we  must  discard  those  that  are 
reducible,  which  presents  no  theoretical  difficulty.  Since 
the  number  of  equations  corresponding  to  a  given  value  of 


TRANSCENDENTAL  NUMBERS.  53 

N  is  limited,  there  corresponds  to  a  determinate  N  only  a 
finite  mass  of  algebraic  numbers.  We  shall  designate  this 
by  ^(N).  The  table  contains  the  values  of  </»(!),  <^(2),  ^(3), 
^  (4),  and  the  corresponding  algebraic  numbers  w. 

We  arrange  now  the  algebraic  numbers  according  to  their 
height,  N,  and  the  numbers  corresponding  to  a  single  value  of 
N  in  increasing  magnitude.  We  thus  obtain  all  the  algebraic 
numbers,  each  in  a  determinate  place.  This  is  done  in  the 
last  column  of  the  accompanying  table.  It  is,  therefore, 
evident  that  algebraic  numbers  can  be  counted. 

3.     We  now  state  the  general  proposition : 

In  any  portion  of  the  axis  of  abscissas,  however  small,  there 
is  an  infinite  number  of  points  which  certainly  do  not  belong  to 
a  given  countable  ma^s. 

Or,  in  other  words  : 

The  continuum  of  num,erical  values  represented  by  a  portion 
of  the  axis  of  abscissas,  however  small,  has  a  greater  power 
than  any  given  countable  mass. 

This  amounts  to  affirming  the  existence  of  transcendental 
numbers.  It  is  sufficient  to  take  as  the  countable  mass  the 
aggregate  of  algebraic  numbers. 

To  demonstrate  this  theorem  we  prepare  a  table  of  algebraic 
numbers  as  before  and  write  in  it  all  the  numbers  in  the  form 
of  decimal  fractions.  Xone  of  these  will  end  in  an  infinite 
series  of  9's.     For  the  equality 

1  =  0.999  •  .  .  9  •  •  • 

shows  that  such  a  number  is  an  exact  decimal.  If  now  we 
can  construct  a  decimal  fraction  which  is  not  found  in  our 
table  and  does  not  end  in  an  infinite  series  of  9's  it  will 
certainly  be  a  transcendental  number.  By  means  of  a  very 
simple  process  indicated  by  Georg  Cantor  we  can  find  not 
only  one  but  infinitely  many  transcendental  numbers,  even 


54  FAMOUS  PROBLEMS. 

when  the  domain  in  which  the  number  is  to  lie  is  very  small. 
Suppose,  for  example,  that  the  first  five  decimals  of  the  num- 
ber are  given.     Cantor's  process  is  as  follows. 

Take  for  6th  decimal  a  number  different  from  9  and  from 
the  6th  decimal  of  the  first  algebraic  number,  for  7th  decimal 
a  number  different  from  9  and  from  the  7th  decimal  of  the 
second  algebraic  number,  etc.  In  this  way  we  obtain  a  decimal 
fraction  which  will  not  end  in  an  infinite  series  of  9's  and  is 
certainly  not  contained  in  our  table.  The  proposition  is  then 
demonstrated. 

We  see  by  this  that  (if  the  expression  is  allowable)  there 
are  far  more  transcendental  numbers  than  algebraic.  For 
when  we  determine  the  unknown  decimals,  avoiding  the  9's, 
we  have  a  choice  among  eight  different  numbers ;  we  can 
thus  form,  so  to  speak,  8*  transcendental  numbers,  even  when 
the  domain  in  which  they  are  to  lie  is  as  small  as  we  please. 


CHAPTER   II. 

Historical  Survey  of  the  Attempts  at  the  Computation 
and  Construction  of  it. 

In  the  next  chapter  we  shall  prove  that  the  number  ir 
belongs  to  the  class  of  transcendental  numbers  whose  exis- 
tence was  shown  in  the  preceding  chapter.  The  proof  was 
first  given  by  Lindemann  in  1882,  and  thus  a  problem  was 
definitely  settled  which,  so  far  as  our  knowledge  goes,  has 
occupied  the  attention  of  mathematicians  for  nearly  4000 
years,  the  problem  of  the  quadrature  of  the  circle. 

For,  if  the  number  tt  is  not  algebraic,  it  certainly  cannot 
be  constructed  by  means  of  straight  edge  and  compasses. 
The  quadrature  of  the  circle  in  the  sense  understood  by  the 
ancients  is  then  impossible.  It  is  extremely  interesting  to 
follow  the  fortunes  of  this  problem  in  the  various  epochs  of 
science,  as  ever  new  attempts  were  made  to  find  a  solution 
with  straight  edge  and  compasses,  and  to  see  how  these  neces- 
sarily fruitless  efforts  worked  for  advancement  in  the  mani- 
fold realm  of  mathematics. 

The  following  brief  historical  survey  is  based  upon  the 
excellent  work  of  Rudio :  Archimedes,  Huygens,  Lambert, 
Lerjendre,  Vier  Abhandlungen  iiber  die  Kreismessung,  Leipzig, 
1892.  This  book  contains  a  German  translation  of  the 
investigations  of  the  authors  named.  While  the  mode  of 
presentation  does  not  touch  upon  the  modern  methods  here 
discussed,  the  book  includes  many  interesting  details  which 
are  of  practical  value  in  elementary  teaching. 


65  FAMOUS  PROBLEMS. 

1.  Among  the  attempts  to  determine  the  ratio  of  the 
diameter  to  the  circumference  we  may  first  distinguish  the 
empirical  stage,  in  which  the  desired  end  was  to  be  attained  by 
measurement  or  by  direct  estimation. 

The  oldest  known  mathematical  document,  the  Ehind 
Papyrus  (c.  2000  b.c),  contains  the  problem  in  the  well- 
known  form,  to  transform  a  circle  into  a  square  of  equal 
area.  The  writer  of  the  papyrus,  Ahmes,  lays  down  the 
following  rule :  Cut  off  ^  of  a  diameter  and  construct  a 
square  upon  the  remainder ;  this  has  the  same  area  as  the 
circle.  The  value  of  tt  thus  obtained  is  (^^Y  =  3.16  •  •  •,  not 
very  inaccurate.  Much  less  accurate  is  the  value  tt  ^  3, 
used  in  the  Bible  (1  Kings,  7.  23,   2  Chronicles,  4.  2). 

2.  The  Greeks  rose  above  this  empirical  standpoint,  and 
especially  Archimedes,  who,  in  his  work  kvkXov  fiiTp-qais,  com- 
puted the  area  of  the  circle  by  the  aid  of  inscribed  and  cir- 
cumscribed polygons,  as  is  still  done  in  the  schools.  His 
method  remained  in  use  till  the  invention  of  the  differential 
calculus  ;  it  was  especially  developed  and  rendered  practical 
by  Huygens  (d.  1654)  in  his  work,  De  circuli  magnitvdine 
inventa. 

As  in  the  case  of  the  duplication  of  the  cube  and  the 
trisection  of  the  angle  the  Greeks  sought  also  to  effect  the 
quadrature  of  the  circle  by  the  help  of  higher  curves. 

Consider  for  example  the  curve  y  =  sin~*  x,  which  repre- 
sents the  sinusoid  with  axis  vertical.  Geometrically,  tt 
appears  as  a  particular  ordinate  of  this  curve ;  from  the 
standpoint  of  the  theory  of  functions,  as  a  particular  value  of 
our  transcendental  function.  Any  apparatus  which  describes 
a  transcendental  curve  we  shall  call  a  transcendental  appara- 
tus. A  transcendental  apparatus  which  traces  the  sinusoid 
gives  us  a  geometric  construction  of  tt. 

In   modern   language   the   curve   y  =  sin"'  x    is    called  an 


THE   CONSTBUCTION  OF  ic. 


67 


integral  curve  because   it  can  be  defined  by   means   of  the 
integral  of  an  algebraic  function, 

dx 


/ax 


Fig.  14. 


The  ancients  called  such  a  curve  a  quadratrix  or  TCTpuywvt- 
^ouo-a.  The  best  known  is  the  quadratrix  of  Dinostratus 
(c.  350  B.C.)  which,  however,  had  al- 
ready been  constructed  by  Hippias  of 
Elis  (c.  420  B.C.)  for  the  trisection  of 
an  angle.  Geometrically  it  may  be 
defined  as  follows.  Having  given  a 
circle  and  two  perpendicular  radii  OA 
and  0  B,  two  points  M  and  L  move  with 
constant  velocity,  one  upon  the  radius 
OB,  the  other  upon  the  arc  AB  (Fig. 
14).  Starting  at  the  same  time  at  0 
and  A,  they  arrive  simultaneously  at  B.  The  point  of  inter- 
section P  of  OL  and  the  parallel  to  OA  through  M  describes 
the  quadratrix. 

From  this  definition  it  follows  that  y  is  proportional  to  6. 

IT 

Further,  since  for  y  ^  1,  ^  =  ;r-  we  have 

and  from  ^  =  tan~^-  the  equation  of  the  curve  becomes 

X 

y  TT 

=^=tan-y. 
X  I  ^ 

It  meets  the  axis  of  X  at  the  point  whose  abscissa  is 

y 

X  =  lim  — - — ,  for  y  =  0 ; 
tan|y 


68 


FAMOUS  PROBLEMS. 


hence 


2 

x  =  — • 

TT 


According  to  this  formula  the  radius  of  the  circle  is  the 
mean  proportional  between  the  length  of  the  quadrant  and 
the  abscissa  of  the  intersection  of  the  quadratrix  with  the 
axis  of  X.  This  curve  can  therefore  be  used  for  the  rectifica- 
tion and  hence  also  for  the  quadrature  of  the  circle.  This 
use  of  the  quadratrix  amounts,  however,  simply  to  a  geo- 
metric formulation  of  the  problem  of  rectification  so  long  as 
we  have  no  apparatus  for  describing  the  curve  by  continuous 
movement. 

Fig.  15  gives  an  idea  of  the  form  of  the  curve  with  the 
branches  obtained  by  taking  values  of  6  greater  than  tt  or 


Fig.  15. 

less  than  — tt.  Evidently  the  quadratrix  of  Dinostratus  is 
not  so  convenient  as  the  curve  y  =  sin~^  x,  but  it  does  not 
appear  that  the  latter  was  used  by  the  ancients. 

3.  The  period  from  1670  to  1770,  characterized  by  the 
names  of  Leibnitz,  Newton,  and  Euler,  saw  the  rise  of  modern 
analysis.  Great  discoveries  followed  one  another  in  such  an 
almost  unbroken  series  that,  as  was  natural,  critical  rigor  fell 
into  the  background.      For  our  purposes   the  development 


THE   COMPUTATION   OF  n.  69 

of  the  theory  of  series  is  especially  important.  Numerous 
methods  were  deduced  for  approximating  the  value  of  tt.  It 
will  suffice  to  mention  the  so-called  Leibnitz  series  (known, 
however,  before  Leibnitz) : 

This  same  period  brings  the  discovery  of  the  mutual  depend- 
ence of  e  and  tt.  The  number  e,  natural  logarithms,  and 
hence  the  exponential  function,  are  first  found  in  principle  in 
the  works  of  Napier  (1614).  This  number  seemed  at  first  to 
have  no  relation  whatever  to  the  circular  functions  and  the 
number  tt  until  Euler  had  the  courage  to  make  use  of  imagi- 
nary exponents.  In  this  way  he  arrived  at  the  celebrated 
formula 

e"  =  cos  X  +  i  sin  x, 
which,  for  x  =  tt,  becomes 

e*'^  =  —  1. 

This  formula  is  certainly  one  of  the  most  remarkable  in  all 
mathematics.  The  modern  proofs  of  the  transcendence  of  tt 
are  all  based  on  it,  since  the  first  step  is  always  to  show  the 
transcendence  of  e. 

4.  After  1770  critical  rigor  gradually  began  to  resume  its 
rightful  place.  In  this  year  appeared  the  work  of  Lambert : 
Vorldufige  Kenntnisse  fur  die  so  die  Quadratur  des  Cirkuls 
suchen.  Among  other  matters  the  irrationality  of  tt  is  dis- 
cussed.  In  1794  Legendre,  in  his  Elements  de  geometrie, 
showed  conclusively  that  tt  and  tt^  are  irrational  numbers. 

5.  But  a  whole  century  elapsed  before  the  question  was 
investigated  from  the  modern  point  of  view.  The  starting- 
point  was  the  work  of  Herraite  :  Sur  la  fonction  exponentieUe 
(Comptes  rendus,  1873,  published  separately  in  1874).  The 
transcendence  of  e  is  here  proved. 


60  FAMOUS  PROBLEMS. 

An  analogous  proof  for  tt,  closely  related  to  that  of 
Hermite,  was  given  by  Lindemann :  Ueber  die  Zahl  tr 
(Mathematische  Annalen,  XX,  1882.  See  also  the  Proceed- 
ings of  the  Berlin  and  Paris  academies). 

The  question  was  then  settled  for  the  first  time,  but  the 
investigations  of  Hermite  and  Lindemann  were  still  very 
complicated. 

The  first  simplification  was  given  by  Weierstrass  in  the 
Berliner  Berichte  of  1885.  The  works  previously  mentioned 
were  embodied  by  Bachmann  in  his  text-book,  Vorlesungen 
uber  die  Natur  der  Irrationalzahlen,  1892. 

But  the  spring  of  1893  brought  new  and  very  important 
simplifications.  In  the  first  rank  should  be  named  the 
memoirs  of  Hilbert  in  the  Gottinger  Nachrichten.  Still 
Hubert's  proof  is  not  absolutely  elementary  :  there  remain 
traces  of  Hermite's  reasoning  in  the  use  of  the  integral 

0 

But  Hurwitz  and  Gordan  soon  showed  that  this  transcen- 
dental formula  could  be  done  away  with  {Gottinger  Nach- 
richten; Comptes  rendus ;  all  three  papers  are  reproduced 
with  some  extensions  in  Mathematische  Annalen,  Vol.  XLIII). 
The  demonstration  has  now  taken  a  form  so  elementary 
that  it  seems  generally  available.  In  substance  we  shall 
follow  Gordan's  mode  of  treatment. 


CHAPTER    III. 

The  Transcendence  of  the  Number  e. 

1.  We  take  as  the  starting-point  for  our  investigation  the 
well-known  series 

X         x^  x° 

which  is  convergent  for  all  finite  values  of  x.     The  difference 

between  practical  and  theoretical  convergence  should  here  be 

insisted  on.      Thus,  for  x  ^  1000  the  calculation  of  e'*^  by 

means  of  this  series  would  obviously  not  be  feasible.     Still 

the  series  certainly  converges   theoretically  ;    for  we  easily 

see   that   after   the    1000th    term   the   factorial    n !    in   the 

denominator  increases  more  rapidly  than  the  power  which 

x° 
occurs  in  the  numerator.     This  circumstance  that  — -  has  for 

n! 

any  finite  value  of  x  the  limit  zero  when  n  becomes  infinite 

has  an  important  bearing  upon  our  later  demonstrations. 

We  now  propose  to  establish  the  following  proposition : 

The  number  e  is  not  an  algebraic  number,  i.e.,  .an  equation 

with  integral  coefficients  of  the  form 

F(e)=Co  +  Cie  +  C2e=^  +  -  '  •  +  C„e"  =  0 

is  impossible.     The  coefficients  Q  may  be  supposed  prime  to 
one  another. 

We  shall  use  the  indirect  method  of  demonstration,  show- 
ing that  the  assumption  of  the  above  equation  leads  to  an 
absurdity.     The  absurdity  may  be  shown  in  the  following 


62  FAMOUS  PROBLEMS. 

way.    We  multiply  the  members  of  the  equation  F(e)  =  0  by 
a  certain  integer  M  so  that 

MF(e)=MCo+MCie+MC2e2+-  •  •+MC„e"  =  0. 

We  shall  show  that  the  number  M  can  be  chosen  so  that 

(1)  Each  of  the  products  Me,  Me^  •  •  -Me"  may  be  sepa- 
rated into  an  entire  part  M^  and  a  fractional  part  c^,  and  our 
equation  takes  the  form 

M  F(e)  =  MCo  +  MiCi  +  M2C2  +  '  •  •  +  M„C„ 

+  Cici    +  0262    +  •  •  ■  +  C„£„  =  0 ; 

(2)  The  integral  part 

MCo+MA  +  -  •  •+M„C„ 
is  not  zero.     This  will  result  from  the  fact  that  when  divided 
by  a  prime  number  it  gives  a  remainder  different  from  zero ; 

(3)  The  expression 

Cici  -f  0262  +  •  •  •  +  C„c„ 

can  be  made  as  small  a  fraction  as  we  please. 

These  conditions  being  fulfilled,  the  equation  assumed  is 
manifestly  impossible,  since  the  sum  of  an  integer  different 
from  zero,  and  a  proper  fraction,  cannot  equal  zero. 

The  salient  point  of  the  proof  may  be  stated,  though  not 
quite  accurately,  as  follows : 

With  an  exceedingly  small  error  we  may  assume  e,  eV  '  '  e" 
proportional  to  integers  which  certainly  do  not  satisfy  our 
assumed  equation. 

2.  We  shall  make  use  in  our  proof  of  a  symbol  h'  and  a 
certain  polynomial  «^(x). 

The  symbol  h'  is  simply  another  notation  for  the  factorial  r ! 
Thus,  we  shall  write  the  series  for  e'  in  the  form 

^'  =  i  +  h  +  P  +  ---  +  ^  +  ---       ■ 


TRANSCENDENCE   OF   THE  NUMBER  e.  63 

The  symbol  has  no  deeper  meaning  ;  it  simply  enables  us  to 
write  in  more  compact  form  every  formula  containing  powers 
and.  factorials. 

Suppose,  e.g.,  we  have  given  a  developed  polynomial 

r 

We  represent  by  f(h),  and  write  under  the  form  2  Crh',  the 
sum 

Ci-l  +  C2-2!  +  C3-3!+-  •  •  +  c„-n! 

But  if  f  (x)  is  not  developed,  then  to  calculate  f  (h)  is  to 
develop  this  polynomial  in  powers  of  h  and  finally  replace 
h'  by  r !.     Thus,  for  example, 

f(k+h)  =  Xc,(k  +  hy  =  2:c','h'  =  Sc',T!, 

r  r  r 

the  c'r  depending  on  k. 

The  polynomial  ^(x)  which  we  need  for  our  proof  is  the 
following  remarkable  expression 

,r(l  — x)(2  — x)-  •  -(n-x)]" 
^ (x)  =  x^i L_i JK  ^^^^^^    <. ZJ_, 

where  p  is  a  prime  number,  n  the  degree  of  the  algebraic 
equation  assumed  to  be  satisfied  by  e.  We  shall  suppose  p 
greater  than  n  and  |Co|,  and  later  we  shall  make  it  increase 
without  limit. 

To  get  a  geometric  picture  of  this  polynomial  <{>  (x)  we  con- 
struct the  curve 

y  =  <^(x). 

At  the  points  x  =  1,  2,  •  •  •  n  the  curve  has  the  axis  of  X  as 
an  inflexional  tangent,  since  it  meets  it  in  an  odd  number  of 
points,  while  at  the  origin  the  axis  of  X  is  tangent  without 
inflexion.  For  values  of  x  between  0  and  n  the  curve  remains 
in  the  neighborhood  of  the  axis  of  X  ;  for  greater  values  of  x 
it  recedes  indefinitely. 


g4  FAMOUS  PROBLEMS. 

Of  the  function  <^  (x)  we  will  now  establish  three  important 
properties : 

1.  X  being  supposed  given  and  p  increasing  without  limit, 
<j>(x)  tends  toward  zero,  as  does  also  the  sum  of  the  absolute 
values  of  its  terms. 

Put  u  =  X  (1  —  x)  (2  —  x)  •  •  •  (n  —  x)  ;  we  may  then  write 

uP-i        u 

</>(x)  =  (^z:i)T  r 

which  for  p  infinite  tends  toward  zero. 

To  have  the  sum  of  the  absolute  values  of  «^  (x)  it  is  suffi- 
cient to  replace  —  x  by  |x|  in  the  undeveloped  form  of  </)(x). 
The  second  part  is  then  demonstrated  like  the  first. 

2.  h  being  an  integer,  <f)  (h)  is  an  integer  not  divisible  by  p 
and  therefore  different  from  zero. 

Develop  ^(x)  in  increasing  powers  of  x,  noticing  that  the 
terms  of  lowest  and  highest  degree  respectively  are  of  degree 
p  —  1  and  np  +  p  —  1.     We  have 

r=np+p — 1 

I5»P— 1 

Hence 

r=np+p— 1 

«^(h)=S    C,h'. 
r=p-l 

Leaving  out  of  account  the  denominator  (p  —  1) !,  which 
occurs  in  all  the  terms,  the  coefficients  c^  are  integers.  This 
denominator  disappears  as  soon  as  we  replace  h""  by  r!,  since 
the  factorial  of  least  degree  is  h^-^  =  (p  —  1) !,  All  the  terms 
of  the  development  after  the  first  will  contain  the  factor  p. 
As  to  the  first,  it  may  be  written 

(l-2-3---n)--(p-l)!^ 

(p-1)!  ^"••' 

and  is  certainly  not  divisible  by  p  since  p  >  n. 
Therefore  ^  (h)  =  (n  !)p  (mod.  p), 

and  hence  <^(h)p£0. 


TRANSCENDENCE   OF   THE  NUMBER  e.  65 

Moreover,  <fi  (h)  is  a  veiy  large  number ;  even  its  last  term 
alone  is  very  large,  viz.: 

V-y=p(p+i)---("p+p-i)- 

3.    h  being  an  integer,  and  k  one  of  the  numbers  1,  2  •  •  •  n, 
<^  (h  +  k)  is  an  integer  divisible  by  p. 

We  have      <^(h  +  k)  =  Sc,(h  +  ky=^c',h', 

r  r 

a  formula  in  which  we  are  to  replace  h''  by  r !  only  after  hav- 
ing arranged  the  development  in  increasing  powers  of  h. 

According  to  the  rules  of  the  symbolic  calculus,  we  have 
first 

_  r(i-k-h)(2-k-h)-  •  -(-h)-  •  ■(n-k-h)r 

"^""^•"^  (P-I)! 

One  of  the  factors  in  the  brackets  reduces  to  —  h  ;  hence  the 
term  of  lowest  degree  in  h  in  the  development  is  of  degree  p. 
We  may  then  write 

r=np+p — 1 

^(h  +  k)  =  Sc',h'. 

r=p 

The  coefficients  still  have  for  numerators  integers  and  for 
denominator  (p  —  1) ! .  As  already  explained,  this  denomi- 
nator disappears  when  we  replace  h'  by  r!.  But  now  all  the 
terms  of  the  development  are  divisible  by  pj  for  the  first 
may  be  written 

(—  1)^P  •  kP-^  [(k  —  1) !  (n  —  k)  !]p  •  p !  ~ 

(p-1)! 

=  (-  1)^P  V^^  [(k  -  1) !  •  (n  —  k)  !]P  •  p. 

^  (h  -f-  k)  is  then  divisible  by  p. 

3.     We  can  now  show  that  the  equation 

F(e)=Co+Cie  +  C2e-'+-  •  •  +  C„e"  =  0 
is  impossible. 


66  FAMOUS  PROBLEMS. 

For  the  number  M,  by  which  we  multiply  the  members  of 
this  equation,  we  select  <f>  (h),  so  that 
</►  (h)  F  (e)  =  Co«^  (h)  -f  Ci<^  (h)  e  +  C2<^  (h)  e^'  + h  C„<^  (h)e". 

Let  us  try  to  decompose  any  term,  such  as  C^^  (h)  e'',  into  an 
integer  and  a  fraction.     We  have 

T 

Considering  the  series  development  of  e^,  any  term  of  this 
sum,  omitting  the  constant  coefficient,  has  the  form 

,     ,   h'  •  k   ,   h'  •  k^  ,   h'  •  k'  .    h^  •  k'+i  . 


2!      '  '      r!      '   (r+1)!   ' 

Keplacing  h"^  by  r !,  or  what  amounts  to  the  same  thing,  by  one 
of  the  quantities 

r\\^\  r(r-l)h'-2-  •  •,  r(r-l)-  •  •3•h^  r(r-l)-  •  •2-h, 
and  simplifying  the  successive  fractions, 

e^  •  h--  =  h^  +  J  •  h'-^k  +  '  ^'~  ^^  h'-^k"  +  •  •  •-f-jhk'-i+k' 

[k  k^  ~| 

7+T+(r  +  l)(rH-2)  +  '  ••]• 

The  first  line  has  the  same  form  as  the  development  of 
(h  +  k)' ;  in  the  parenthesis  of  the  second  line  we  have  the 
series 

k  k^ 

^r  +  l^(r  +  l)(r  +  2)^ 

whose  terms  are  respectively  less  than  those  of  the  series 

k^       k' 
e'  =  l  +  k  +  |j  +  |j+--- 

The  second  line  in  the  expansion  of  e*"  •  h""  may  therefore  be 
represented  by  an  expression  of  the  form 

qr.ic  •  e^  •  k', 
q,^k  being  a  proper  fraction. 


TRANSCENDENCE   OF   THE  NUMBER  e.  67 

Effecting  the  same  decomposition  for  each  term  of  the  sum 

r 

it  takes  the  form 

e''  t  oy  =  S  c,  (h  +  k)'  +e^%  qr.kCrk'. 

r  r  r 

The  first  part  of  this  sum  is  simply  ^(h  +  k);  this  is  a 
number  divisible  by  p  (2,  3).     Further  (2,  1), 

<^(k)  =  ^|c,k^| 

I  I       I 

tends  toward  zero  when  p  becomes  infinite  :   the  same  is  true 
a  fortiori  of  Xqr.kCrk'j  and  also,  since  e''  is  a  finite  quantity, 

r 

of  e^'^q^kCrk"',  which  we  may  represent  by  c^. 

r 

The  term  under  consideration,  Cke^^  (h),  has  then  been  put 
under  the  form  of  an  integer  Ck<^(h  +  k)  and  a  quantity  C^e^ 
which,  by  a  suitable  choice  of  p,  may  be  made  as  small  as  we 
please. 

Proceeding  similarly  with  all  the  terms,  we  get  finally 
F(e)<^(h)  =  Co<^(h)  +  Ci<^(h+l)+-  •  •  +  C„<^(h  +  n) 

+  CiCi  +  0262  +  •    •    •  +  Cne„. 

It  is  now  easy  to  complete  the  demonstration.  All  the 
terms  of  the  first  line  after  the  first  are  divisible  by  p  ;  for 
the  first,  I  Co  I  is  less  than  p  ;  ^  (h)  is  not  divisible  by  p;  hence 
Co<^(h)  is  not  divisible  by  the  prime  number  p.  Consequently 
the  sum  of  the  numbers  of  the  first  line  is  not  zero. 

The  numbers  of  the  second  line  are  finite  in  number ;  each 
of  them  can  be  made  smaller  than  any  given  number  by  a 
suitable  choice  of  p  ;  and  therefore  the  same  is  true  of  their 
sum. 

Since  an  integer  not  zero  and  a  fraction  cannot  have  zero 
for  a  sum,  the  assumed  equation  is  impossible. 

Thus,  the  transcendence  of  e,  or  Hermite's  Theorem,  is 
demonstrated. 


CHAPTER    IV. 

The  Transcendence  of  the  Number  it. 

1.  The  demonstration  of  the  transcendence  of  the  number 
TT  given  by  Lindemann  is  an  extension  of  Hermite's  proof  in 
the  case  of  e.  While  Hermite  shows  that  an  integral  equa- 
tion  of  the  form 

Co  +  Cie  +  C2e^+-  •  •  +  C„e"  =  0 
cannot  exist,  Lindemann  generalizes  this  by  introducing  in 
place  of  the  powers  e,  e'^  •  •  •  sums  of  the  form 

e  1  +  e  2  +  •  •  •  +  e'"' 


where  the  k's  are  associated  algebraic  numbers,  i.e.,  roots  of 
an  algebraic  equation,  with  integral  coefficients,  of  the  degree 
N  ;  the  I's  roots  of  an  equation  of  degree  N',  etc.  Moreover, 
some  or  all  of  these  roots  may  be  imaginary. 

Lindemann's  general  theorem  may  be  stated  as  follows : 

The  number  e  cannot  satisfy  an  equation  of  the  form 
(1)    Co+Ci(e''»  +  e'2+-  •  •  +  e'''') 

+  C,(e'^+e^^+-  •  •  +  e''')+-  •  -  =  0 
where  the  coefficients  Q  are  integers  and  the  exponents  kj,  Ij,  •  •  • 
are  respectively  associated  algebraic  numbers. 

The  theorem  may  also  be  stated  : 

The  number  e  is  not  only  not  an  algebraic  numher  and  there- 
fore a  transcendental  number  simply,  but  it  is  also  not  an 
interscendental  *  number  and  therefore  a  transcendental  number 
of  higher  order. 

*  Leibnitz  calls  a  function  x^,  where  \  is  an  algebraic  irrational,  an 
interscendental  function. 


TRANSCENDENCE   OF   THE  NUMBER   it.  69 

Let 

ax"  +  aix''"^  +  •  •  •  +  Bj,  =  0  ' 

be  the  equation  having  for  roots  the  exponents  kj ; 

bx'' -f- bix"'-i  +  •  •  •  +  b^.  =  0 
that  having  for  roots  the  exponents  l^,  etc.  These  equations 
are  not  necessarily  irreducible,  nor  the  coefficients  of  the  first 
terms  equal  to  1.  It  follows  that  the  symmetric  functions  of 
the  roots  which  alone  occur  in  our  later  developments  need 
not  be  integers. 

In  order  to  obtain  integral  numbers  it  will  be  sufficient  to 
consider  symmetric  functions  of  the  quantities 

aki,  aK2)  *  *   ■  aKjfj 
blj,  biz,   •  •  •  bl^.,  etc. 

These  numbers  are  roots  of  the  equations 

y"  +  aiy"-^  +  ajay''-^  +  .••-(-  a^a"-^  =  0, 
y"'  +  biy"'-^  +  b2by»'-2  +  •  •  •  +  b^.b^'-^  =  0,  etc. 

These  quantities  are  integral  associated  algebraic  numbers, 
and  their  rational  symmetric  functions  real  integers. 

We  shall  now  follow  the  same  course  as  in  the  demonstra- 
tion of  Hermite's  theorem. 

We  assume  equation  (1)  to  be  true  ;  we  multiply  both 
members  by  an  integer  M ;  and  we  decompose  each  sum, 
such  as 

M(e'^^  +  e''=^+-  •  -H-e^"), 
into  an  integral  part  and  a  fraction,  thus 

M  (e'^i  +  e'^^  +  •  •  •  +  e''0==Mi  +  ci, 
M(e'i  +  e'2  +  -  •  •+eO=M2  +  ea, 


Our  equation  then  becomes 

+  Cici   +C,c,  +•  •  -  =  0. 


70  FAMOUS  PROBLEMS. 

We  shall  show  that  with  a  suitable  choice  of  M  the  sum  of 
the  quantities  in  the  first  line  represents  an  integer  not 
divisible  by  a  certain  prime  number  p,  and  consequently- 
different  from  zero  ;  that  the  fractional  part  can  be  made  as 
small  as  we  please,  and  thus  we  come  upon  the  same  contra- 
diction as  before. 

2.  We  shall  again  use  the  symbol  h'^r!  and  select  as 
the  multiplier  the  quantity  M  =  i/^  (h),  where  ij/  (x)  is  a  gene- 
ralization of  ^(x)  used  in  the  preceding  chapter,  formed  as 
follows  : 


p-i 


•  [(li  -  X)  (I,  -  X)  •  •  •  (I,,  -  x)]P  •  b-P  •  h^'^  •  b-"P  •  •  • 

m 

where  p  is  a  prime  number  greater  than  the  absolute  value  of   ^ 
each  of  the  numbers 

Co,  a?  b,  •  •  ■,  a^,  b^.,  • 
and  later  will  be  assumed  to  increase  without  limit.     As  to 
the  factors  a''^,  b"'",  •  •  •,  they  have  been  introduced  so  as  to 
have  in  the  development  of  i/r(x)  symmetric  functions  of  the 
quantities 

akj,  ak],  •  •  •,  ak^, 
bli,  bla,  •  •  •,  bl,,, 


that  is,  rational  integral  numbers.  Later  on  we  shall  have 
to  develop  the  expressions 

V  V 

The  presence  of  these  same  factors  will  still  be  necessary  if 
we  wish  the  coefficients  of  these  developments  to  be  integers 
each  divided  by  (p  —  1) !. 

1.    i/'(h)  is  an  integral  number,  not  divisible  hy  p  and  con- 
se^uently  different  from  zero. 


TRANSCENDENCE   OF   THE  NUMBER   it.  71 

Arranging  \p  (h)  in  increasing  powers  of  h,  it  takes  the  form 

r=»p+K'p+  •  •  •  +p— 1 
r=p — 1 

In  this  development  all  the  coefficients  have  integral  numer- 
ators and  the  common  denominator  (p  —  1) !. 

The  coefficient  of  the  first  term  h^"'  may  be  written 

_^     (aki  •  aka  •  •   •  ak.,)Pa'''Pa''"P  •  •  • 
•  (bli  •  bla  •  •  •  bL,.)Pb''Pb""p  •  •  • 


.(_l-)KP+>rp+ •••(a^a''-^)Pa'''Pa'''p-  •  •(b„.b'''-^)Pb''Pb'''P 


(p-i)r 

If  in  this  term  we  replace  h^"^  by  its  value  (p  — 1)!  the 
denominator  disappears.  According  to  the  hypotheses  made 
regarding  the  prime  number  p,  no  factor  of  the  product  is 
divisible  by  p  and  hence  the  product  is  not. 

The  second  term  Cph^  becomes  likewise  an  integer  when 
we  replace  h^  by  p!  but  the  factor  p  remains,  and  so  for  all 
of  the  following  terms.  Hence  i/^(h)  is  an  integer  not  divis- 
ible by  p. 

2.  For  X,  a  given  finite  quantity,  and  p  increasing  without 
limit,   i/'  (x)  =  X  CrX""  tends  toward  zero,  as  does  also   the  sum 

SI  CXI- 

r 

We  may  write 

=  -(^^ll.^^^"  ■  •  •b-b'''(l<i-x)(k,-x)-  •  -(k.-x) 

(l-x)(l2-x)-  •  •(I..-X)-  •  -p. 
Since  for  x  of  given  value  the  expression  in  brackets  is  a  con- 
stant, we  may  replace  it  by  K.     We  then  have 

(xK)P-'  „ 

a  quantity  which  tends  toward  zero  as  p  increases  indefinitely. 


72  FAMOUS  PROBLEMS. 

The  same  reasoning  will  apply  when  each  term  of  i/^(x)  is 
replaced  by  its  absolute  value. 

3.    The  repression  ^  (/^  (k^,  +  h)  is  an  integer  divisihle  hy  p. 

We  have 

-A^+h)  =  ^^^^^i^b''Pb-P.-  • 

•a('-^>''[(kx-k,-h)(k2-k,-h)-  •  -(-h)-  •  -(k.-k.-h)]" 
•a'''Pb''P[(li-k,-h)(l2-k,-h)-  •  •(!. -k,-h)]p 

The  i/th  factor  of  the  expression  in  brackets  in  the  second 
line  is  —  h,  and  hence  the  term  of  lowest  degree  in  h  is  h^. 
Consequently 

r=Np+N'p+  •  •  •  +p— 1 

,^(k,+h)=    X    c'M 
i=p 

whence 

V=lf  I=Np+»'p+  •  •  •  +P— 1 

Xvr(k,+h)=   t   cy. 

v=\  r=p 

The  numerators  of  the  coefficients  C,  are  rational  and  integral, 
for  they  are  integral  symmetric  functions  of  the  quantities 

aki,         ak2,         •  ■  *j         3k^> 
bli,         biz,         •  •  •,         bij,.. 


and  their  common  denominator  is  (p  —  1) ! . 

If  we  replace  h""  by  r !  the  denominator  disappears  from  all 
the  coefficients,  the  factor  p  remains  in  every  term,  and  hence 
the  sum  is  an  integer  divisible  by  p. 

Similarly  for 

1/=! 

We  have  thus  established  three  properties  of  \f/  (x)  analogous 
to  those  demonstrated  for  <f>  (x)  in  connection  with  Hermite's 
theorem. 


TEANSC£NDENCE   OF   THE  NUMBER  it.  73 

3.     We  now  return  to  our  demonstration  that  the  assumed 
equation 

(1)  Co+Cx(e'^+e'-^+-  •  •  +  e'--')+C2(ei  +  e'^+-  •  •e'-"')+-  •  -=0 
caunot  be  true.  For  this  purpose  we  multiply  both  members 
by  i/'Ch),  thus  obtaining 

Co./'(h)  +  Ci[e''V(h)  +  e''V(h)  +  -  •  .  +  e'''V'(h)]  +  -  •  -  =  0, 
and  try  to  decompose  each  of  the  expressions  in  brackets  into 
a  whole  number  and  a  fraction.  The  operation  will  be  a  little 
longer  than  before,  for  k  may  be  a  complex  number  of  the  form 
k  =  k'  +  i k".  We  shall  need  to  introduce  |  k  ]  =  +  Vk'^  +  k"^^- 
One  term  of  the  above  sum  is 

e''  • ,/,  (h)  =  e''  S  Crh'  =  :^  c,  •  e''  •  h^ 

r  r 

The  product  e^  •  h''  may  be  written,  as  shown  before, 

^"^'  =  (^  +  '')'  +  ^{7Tl  +  (r  +  lKr  +  2)+-  ■  1 

The  absolute  value  of  every  term  of  the  series 

k                     k^ 
04-— ^H h-  •  • 

is  less  than  the  absolute  value  of  the  corresponding  term  in 

the  series 

k       k^ 
6^  =  14--  +  —  +  •  •  • 


Hence 


+  — — ^-— — +•  •  •  <ei^' 


r4-l   '    (r  +  l)(r4-2) 
k       .  k'' 


k| 


7+T+(r  +  l)(r  +  2)  +  ---  =  1"'^ 
q^^k  being  a  complex  quantity  whose  absolute  value  is  less 
than  1. 

We  may  then  write 

e''  •  ^  (h)  =  S  c.e^h'  =  :£  c,  (h  +  k)'  +  S  c^q^uk^e'"' 
=  ./^(h  +  k)-fScrqr.kk'^-ei>''. 


74 


FAMOUS  PROBLEMS. 


By  giving  k  in  succession  the  indices  1,  2,  •  •  •  n,  and  form- 
ing the  sum  the  equation  becomes 

e''V(h)  +  e'V(h)+-  •  •+e^''^(h) 

=  iV  (K  +  h)  +1'  ^  e  \K\  t  cXqr,kJ. 

Proceeding  similarly  with  all  the  other  sums,  our  equation 
takes  the  form 

(2)  Co  f  (h)  -I-  Ci 'iV  (k.  +  h)  +  Q  iV  (1,  +  h)  +  •  •  • 


(=1 


+  CiS  Se!Mc,k,q,.k^+  C2  S  t  elV  cj'.,q,,^+  •  •  -=0. 
By  2,  2  we  can  make  ^Icrk""!  as  small  as  we  please  by  taking 

r 

p  sufficiently  great.  Since  | q,,, |  <  1,  this  will  be  true  a  fortiori 
of 

t  Crk'qr.k 

r 

and  hence  also  of 

TSc^k^q^^e'^"!. 

>'=l    r 

Since  the  coefficients  C  are  finite  in  value  and  in  number,  the 
sum  which  occurs  in  the  second  line  of  (2)  can,  by  increasing 
p,  be  made  as  small  as  we  please. 

The  numbers  of  the  first  line  are,  after  the  first,  all  divis- 
ible by  p  (3),  but  the  first  number,  Coi/'(h),  is  not  (1). 
Therefore  the  sum  of  the  numbers  in  the  first  line  is  not 
divisible  by  p  and  hence  is  different  from  zero.  The  sura  of 
an  integer  and  a  fraction  cannot  be  zero.  Hence  equation  (2) 
is  impossible  and  consequently  also  equation  (1).* 

4.  We  now  come  to  a  proposition  more  general  than  the 
preceding,  but  whose  demonstration  is  an  immediate  conse- 

*  The  proof  for  the  more  general  case  where  Co  =  0  may  be  reduced 
to  this  by  multiplication  by  a  suitable  factor,  or  may  be  obtained  directly 
by  a  proper  modification  of  ^  (h). 


TRANSCENDENCE   OF   THE  NUMBER   it.  75 

quence  of  the  latter.     For  this  reason  we  shall  call  it  Linde- 
mann's  corollary. 

The  number  e  cannot  satisfy  an  equation  of  the  form 
(3)    C'o+C>'^>  +  C'.e'>+-  •  -  =  0, 

in  which  the  coefficients  are  integers  even  when  the  exponents 
ki,  lij  ■   ■  '  «'■«  unrelated  algebraic  numbers. 

To  demonstrate  this,  let  kg,  kg,  •  •  •,  k^  be  the  other  roots  of 
the  equation  satisfied  by  ki ;  similarly  for  I2,  I3,  •  •  •,  I,.,  etc. 
Form  all  the  polynomials  which  may  be  deduced  from  (3) 
by  replacing  ki  in  succession  by  the  associated  roots  kg,  •  •  •, 
li  by  the  associated  roots  I2,  •  *  '  Multiplying  the  expres- 
sions thus  formed  we  have  the  product 

a  =  l,  2,  •  '  •,  N 

a,  3, •  •  • 


^=1,2,-  •  •,  N' 


=  Co  +  Ci  (e'l  +  e'2  +  •  •  •  +  e''")  +  C^  (e'^-""'  +  e'^*^'^"  +  •  •  •) 
+  C3(e'i^'^  +  e'i^'2+-  ••)+••• 

In  each  parenthesis  the  exponents  are  formed  symmetrically 
from  the  quantities  k^,  Ij,  •  •  •,  and  are  therefore  roots  of  an 
algebraic  equation  with  integral  coefficients.  Our  product 
comes  under  Lindemann's  theorem ;  hence  it  cannot  be  zero. 
Consequently  none  of  its  factors  can  be  zero  and  the  corollary 
is  demonstrated. 

We  may  now  deduce  a  still  more  general  theorem. 

The  7iumber  e  cannot  satisfy  an  equation  of  the  form 
aj)  +  C<}>e''  +  'CWei-f  •  •  -  =  0 
where  the  coefficients  as  well  as  the  exponents  are  unrelated 
algebraic  numbers. 

For,  let  us  form  all  the  polynomials  which  we  can  deduce 
from  the  preceding  when  for  each  of  the  expressions  0^\  we 
substitute  one  of  the  associated  algebraic  numbers 
r(2)   Q.3)   .  .  .  r*") 


76  FAMOUS  PROBLEMS. 

If  we  multiply  the  polynomials  thus  formed  together  we  get 
the  product 

^a  =  1,  2,  •  •  •,  No 


n     ic(s^  +  cre^  +  c(lV+-  •  -\ 


P=\,2,-  •  •,  Ni 
■y  =  1,  2,  •   •    •,  Na 


=  Co  +  C,e''+C,e^+-  •  • 

+ 

+ ' 

where  the  coeflB.cients  C  are  integral  symmetric  functions  of 
the  quantities 


and  hence  are  rational.  By  the  previous  proof  such  an 
expression  cannot  vanish,  and  we  have  accordingly  Linde- 
mann's  corollary  in  its  most  general  form  : 

The  number  e  cannot  satisfy  an  equation  of  the  form 

Co+Cxe''+C2e^  +  -  •  -  =  0 
where  the  exponents  k,  I,  •  •  -as  well  as  the  coefficients  Co,  Ci, 
•  •  •  are  algebraic  numbers. 

This  may  also  be  stated  as  follows : 

In  an  equation  of  the  form 

Co+Cie''+C2e»+-  •  -  =  0 

the  exponents  and  coefficients  cannot  all  be  algebraic  numbers. 

5.  From  Lindemann's  corollary  we  may  deduce  a  number 
of  interesting  results.  First,  the  transcendence  of  tt  is  an 
immediate  consequence.  For  consider  the  remarkable  equa- 
tion 

1  -I-  e'"  =  0. 


TRANSCENDENCE   OF   THE  NUMBER   x.  77 

The  coefficients  of  this  equation  are  algebraic  ;  hence  the 
exponent  itt  is  not.     Therefore,  tt  is  transcendental. 

6.  Again  consider  the  function  y  =  e^.  "We  know  that 
1  =  e".  This  seems  to  be  contrary  to  our  theorems  about  the 
transcendence  of  e.  This  is  not  the  case,  however.  We 
must  notice  that  the  case  of  the  exponent  0  was  implicitly 
excluded.  For  the  exponent  0  the  function  ^(x)  would  lose 
its  essential  properties  and  obviously  our  conclusions  would 
not  hold. 

Excluding  then  the  special  case  (x  =  0.  y  =  1),  Lindemann's 
corollary  shows  that  in  the  equation  y  =:  e''  or  x  =  log^y,  y  and 
X,  i.e.,  the  number  and  its  natural  logarithm,  cannot  be  alge- 
braic simultaneously.  To  an  algebraic  value  of  x  corresponds 
a  transcendental  value  of  y,  and  conversely.  This  is  certainly 
a  very  remarkable  property. 

If  we  construct  the  curve  y  =  e*  and  mark  all  the  algebraic 
points  of  the  plane,  i.e.,  all  points  whose  coordinates  are  alge- 
braic numbers,  the  curve  passes  among  them  without  meeting 
a  single  one  except  the  point  x  =  0.  y  =  1.  The  theorem  still 
holds  even  when  x  and  y  take  arbitrary  complex  values.  The 
exponential  curve  is  then  transcendental  in  a  far  higher  sense 
than  ordinarily  supposed. 

7.  A  further  consequence  of  Lindemann's  corollary  is  the 
transcendence,  in  the  same  higher  sense,  of  the  function 
y  =  sin~^  X  and  similar  functions. 

The  function  y  =  sin~^  x  is  defined  by  the  equation 

2  ix  =  e^-''  —  e-'-'^. 

We  see,  therefore,  that  here  also  x  and  y  cannot  be  algebraic 
simultaneously,  excluding,  of  course,  the  values  x  =  0.  y  =  0. 
We  may  then  enunciate  the  proposition  in  geometric  form  : 

The  curve  y  =  sin~^  x,  like  the  curve  y  =  e^.  liosses  through 
no  algebraic  point  of  the  plane,  except  x  =  0.  y  =  0. 


CHAPTER   V. 
The  Integraph  and  the  Geometric  Construction  of  tt. 

1.  Lindemaun's  theorem  demonstrates  the  transcendence 
of  TT,  and  thus  is  shown  the  impossibility  of  solving  the  old 
problem  of  the  quadrature  of  the  circle,  not  only  in  the  sense 
understood  by  the  ancients  but  in  a  far  more  general  manner. 
It  is  not  only  impossible  to  construct  tt  with  straight  edge 
and  compasses,  but  there  is  not  even  a  curve  of  higher  order 
defined  by  an  integral  algebraic  equation  for  which  tt  is  the 
ordinate  corresponding  to  a  rational  value  of  the  abscissa. 
An  actual  construction  of  tt  can  then  be  effected  only  by  the 
aid  of  a  transcendental  curve.  If  such  a  construction  is 
desired,  we  must  use  besides  straight  edge  and  compasses 
a  ''  transcendental "  apparatus  which  shall  trace  the  curve  by 
continuous  motion. 

2.  Such  an  apparatus  is  the  integraph,  recently  invented 
and  described  by  a  Russian  engineer,  Abdank-Abakanowicz, 
and  constructed  by  Coradi  of  Zurich. 

This  instrument  enables  us  to  trace  the  integral  curve 
Y=F(x)=/f(x)dx 
when  we  have  given  the  differential  curve 

y  =  f(x). 

For  this  purpose,  we  move  the  link  work  of  the  integraph 
so  that  the  guiding  point  follows  the  differential  curve  ;  the 
tracing  point  will  then  trace  the  integral  curve.  For  a  fuller 
description  of  this  ingenious  instrument  we  refer  to  the 
original  memoir  (in  German,  Teubner,  1889 ;  in  French, 
Gauthier-Villars,  1889). 


GEOMETRIC   CONSTRUCTION  OF  it. 


79 


We  shall  simply  indicate  the  principles  of  its  working. 
For  any  point  (x,  y)  of  the  differential  curve  construct  the 
auxiliary  triangle  having  for  vertices  the  points  (x,  y),  (x,  0), 
(x  —  1,  0);  the  hypotenuse  of  this  right-angled  triangle  makes 
with  the  axis  of  X  an  angle  whose  tangent  =  y. 

Hence,  this  hypotenuse  is  parallel  to  the  tangent  to  the  inte- 
gral curve  at  the  point  (y^,  V)  corresponding  to  the  point  (x,  y). 


Y 

4- 

,«^ 

y 

-^ 

5\ 

r 

/ 

r 

~^ 

X 

0 

^f 

0 

J 

*^    4 

-^ 

Fig.  16. 

The  apparatus  should  be  so  constructed  then  that  the  trac- 
ing point  shall  move  parallel  to  the  variable  direction  of  this 
hypotenuse,  while  the  guiding  point  describes  the  diiferential 
curve.  This  is  effected  by  connecting  the  tracing  point  with 
a  sharp-edged  roller  whose  plane  is  vertical  and  moves  so  as  to 
be  always  parallel  to  this  hypotenuse.  A  weight  presses  this 
roller  firmly  upon  the  paper  so  that  its  point  of  contact  can 
advance  only  in  the  plane  of  the  roller. 

The  practical  object  of  the  integraph  is  the  approximate 
evaluation  of  definite  integrals  ;  for  us  its  application  to  the 
construction  of  tt  is  of  especial  interest. 

3.     Take  for  differential  curve  the  circle 
x2  +  y2  =  r'^  J 


30  FAMOUS  PROBLEMS. 

the  integral  curve  is- then 

2  

Y  =/  Vi^=^^=^clx  =  ^  sin-i-  +  ^  Vr^^^7^ 
•^  2  r      J 

This  curve  consists  of  a  series  of  congruent  branches.  The 
points  where  it  meets  the  axis  of  Y  have  for  ordinates 

0,     ±^,     ••• 
Upon  the  lines  X  :=  ±  r  the  intersections  have  for  ordinates 

„7r  „3  TT 

If  we  make  r  =  1,  the  ordinates  of  these  intersections  will 
determine  the  number  tt  or  its  multiples. 

It  is  worthy  of  notice  that  our  apparatus  enables  us  to 
trace  the  curve  not  in  a  tedious  and  inaccurate  manner,  but 
with  ease  and  sharpness,  especially  if  we  use  a  tracing  pen 
instead  of  a  pencil. 

Thus  we  have  an  actual  constructive  quadrature  of  the 
circle  along  the  lines  laid  down  by  the  ancients,  for  our 
curve  is  only  a  modification  of  the  quadratrix  considered 
by  them. 


ANNOUNCEMENTS 


THE  TEACHING  OF  GEOMETRY 

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showing  their  origin,  the  various  methods  of  treating  them,  and 
their  genuine  applications,  thus  giving  to  the  teacher  exactly 
the  material  needed  to  vitalize  the  work  in  the  high  school. 

Great  care  has  been  taken  in  the  illustrations,  particularly 
with  respect  to  the  applications  of  geometry  to  design,  to  men- 
suration, and  to  such  simple  cases  in  physics  as  are  within  the 
easy  reach  of  the  student. 

The  work  cannot  fail  to  set  the  standard  in  geometry  in  this 
country  for  years  to  come,  and  to  stimulate  teachers  to  the 
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